TPTP Problem File: SLH0340^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Query_Optimization/0006_CostFunctions/prob_00439_019632__14973390_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1464 ( 567 unt; 192 typ; 0 def)
% Number of atoms : 3721 (1704 equ; 0 cnn)
% Maximal formula atoms : 15 ( 2 avg)
% Number of connectives : 12518 ( 466 ~; 70 |; 273 &;10244 @)
% ( 0 <=>;1465 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 19 ( 18 usr)
% Number of type conns : 932 ( 932 >; 0 *; 0 +; 0 <<)
% Number of symbols : 177 ( 174 usr; 15 con; 0-5 aty)
% Number of variables : 3509 ( 226 ^;3180 !; 103 ?;3509 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:05:48.794
%------------------------------------------------------------------------------
% Could-be-implicit typings (18)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Complex__Ocomplex_J_J,type,
set_set_complex: $tType ).
thf(ty_n_t__JoinTree__OjoinTree_It__Complex__Ocomplex_J,type,
joinTree_complex: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__JoinTree__OjoinTree_It__Real__Oreal_J,type,
joinTree_real: $tType ).
thf(ty_n_t__List__Olist_It__Complex__Ocomplex_J,type,
list_complex: $tType ).
thf(ty_n_t__JoinTree__OjoinTree_It__Nat__Onat_J,type,
joinTree_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
set_complex: $tType ).
thf(ty_n_t__List__Olist_It__Real__Oreal_J,type,
list_real: $tType ).
thf(ty_n_t__JoinTree__OjoinTree_Itf__a_J,type,
joinTree_a: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Complex__Ocomplex,type,
complex: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (174)
thf(sy_c_CostFunctions_Oc__IKKBZ_001t__Complex__Ocomplex,type,
c_IKKBZ_complex: ( complex > real > real ) > ( complex > real ) > ( complex > complex > real ) > joinTree_complex > real ).
thf(sy_c_CostFunctions_Oc__IKKBZ_001t__Nat__Onat,type,
c_IKKBZ_nat: ( nat > real > real ) > ( nat > real ) > ( nat > nat > real ) > joinTree_nat > real ).
thf(sy_c_CostFunctions_Oc__IKKBZ_001tf__a,type,
c_IKKBZ_a: ( a > real > real ) > ( a > real ) > ( a > a > real ) > joinTree_a > real ).
thf(sy_c_CostFunctions_Oc__list_001tf__a,type,
c_list_a: ( a > real ) > ( a > real ) > ( a > real ) > a > list_a > real ).
thf(sy_c_CostFunctions_Oc__list_H_001tf__a,type,
c_list_a2: ( a > a > real ) > ( a > real ) > ( list_a > a > real ) > list_a > real ).
thf(sy_c_CostFunctions_Oc__nlj_001t__Complex__Ocomplex,type,
c_nlj_complex: ( complex > real ) > ( complex > complex > real ) > joinTree_complex > real ).
thf(sy_c_CostFunctions_Oc__nlj_001t__Nat__Onat,type,
c_nlj_nat: ( nat > real ) > ( nat > nat > real ) > joinTree_nat > real ).
thf(sy_c_CostFunctions_Oc__nlj_001tf__a,type,
c_nlj_a: ( a > real ) > ( a > a > real ) > joinTree_a > real ).
thf(sy_c_CostFunctions_Oc__out_001tf__a,type,
c_out_a: ( a > real ) > ( a > a > real ) > joinTree_a > real ).
thf(sy_c_CostFunctions_Ocreate__h__list_001tf__a,type,
create_h_list_a: ( list_a > a > real ) > list_a > a > real ).
thf(sy_c_CostFunctions_Osymmetric_001tf__a,type,
symmetric_a: ( joinTree_a > real ) > $o ).
thf(sy_c_CostFunctions_Osymmetric_H_001tf__a,type,
symmetric_a2: ( ( a > real ) > ( a > a > real ) > joinTree_a > real ) > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Complex__Ocomplex,type,
finite3207457112153483333omplex: set_complex > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Real__Oreal,type,
finite_finite_real: set_real > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Complex__Ocomplex_J,type,
finite6551019134538273531omplex: set_set_complex > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Complex__Ocomplex,type,
minus_minus_complex: complex > complex > complex ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex,type,
one_one_complex: complex ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Complex__Ocomplex,type,
plus_plus_complex: complex > complex > complex ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Complex__Ocomplex,type,
times_times_complex: complex > complex > complex ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Complex__Ocomplex,type,
uminus1482373934393186551omplex: complex > complex ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
uminus_uminus_real: real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex,type,
zero_zero_complex: complex ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
groups7754918857620584856omplex: ( complex > complex ) > set_complex > complex ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Nat__Onat,type,
groups5693394587270226106ex_nat: ( complex > nat ) > set_complex > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Real__Oreal,type,
groups5808333547571424918x_real: ( complex > real ) > set_complex > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Complex__Ocomplex,type,
groups2073611262835488442omplex: ( nat > complex ) > set_nat > complex ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Nat__Onat,type,
groups3542108847815614940at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Real__Oreal,type,
groups6591440286371151544t_real: ( nat > real ) > set_nat > real ).
thf(sy_c_Groups__List_Omonoid__mult__class_Oprod__list_001t__Complex__Ocomplex,type,
groups7979759902575632448omplex: list_complex > complex ).
thf(sy_c_Groups__List_Omonoid__mult__class_Oprod__list_001t__Nat__Onat,type,
groups6371653412389394274st_nat: list_nat > nat ).
thf(sy_c_Groups__List_Omonoid__mult__class_Oprod__list_001t__Real__Oreal,type,
groups2776710990603637054t_real: list_real > real ).
thf(sy_c_HOL_Oundefined_001t__JoinTree__OjoinTree_Itf__a_J,type,
undefined_joinTree_a: joinTree_a ).
thf(sy_c_HOL_Oundefined_001t__Real__Oreal,type,
undefined_real: real ).
thf(sy_c_If_001t__Complex__Ocomplex,type,
if_complex: $o > complex > complex > complex ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Real__Oreal,type,
if_real: $o > real > real > real ).
thf(sy_c_JoinTree_Ocard_001t__Complex__Ocomplex,type,
card_complex: ( complex > real ) > ( complex > complex > real ) > joinTree_complex > real ).
thf(sy_c_JoinTree_Ocard_001t__Nat__Onat,type,
card_nat: ( nat > real ) > ( nat > nat > real ) > joinTree_nat > real ).
thf(sy_c_JoinTree_Ocard_001tf__a,type,
card_a: ( a > real ) > ( a > a > real ) > joinTree_a > real ).
thf(sy_c_JoinTree_Ocreate__ldeep_001tf__a,type,
create_ldeep_a: list_a > joinTree_a ).
thf(sy_c_JoinTree_Ocreate__ldeep__rev_001tf__a,type,
create_ldeep_rev_a: list_a > joinTree_a ).
thf(sy_c_JoinTree_Ocreate__ldeep__rev__rel_001tf__a,type,
create4822648437634936863_rel_a: list_a > list_a > $o ).
thf(sy_c_JoinTree_Ocreate__rdeep_001tf__a,type,
create_rdeep_a: list_a > joinTree_a ).
thf(sy_c_JoinTree_Ocreate__rdeep__rel_001tf__a,type,
create_rdeep_rel_a: list_a > list_a > $o ).
thf(sy_c_JoinTree_Odistinct__relations_001t__Complex__Ocomplex,type,
distin1760132951325961430omplex: joinTree_complex > $o ).
thf(sy_c_JoinTree_Odistinct__relations_001t__Nat__Onat,type,
distin9127876880396718072ns_nat: joinTree_nat > $o ).
thf(sy_c_JoinTree_Odistinct__relations_001t__Real__Oreal,type,
distin8564637588129275604s_real: joinTree_real > $o ).
thf(sy_c_JoinTree_Odistinct__relations_001tf__a,type,
distinct_relations_a: joinTree_a > $o ).
thf(sy_c_JoinTree_Ofirst__node_001t__Complex__Ocomplex,type,
first_node_complex: joinTree_complex > complex ).
thf(sy_c_JoinTree_Ofirst__node_001t__Nat__Onat,type,
first_node_nat: joinTree_nat > nat ).
thf(sy_c_JoinTree_Ofirst__node_001tf__a,type,
first_node_a: joinTree_a > a ).
thf(sy_c_JoinTree_Oinorder_001tf__a,type,
inorder_a: joinTree_a > list_a ).
thf(sy_c_JoinTree_OjoinTree_OJoin_001t__Complex__Ocomplex,type,
join_complex: joinTree_complex > joinTree_complex > joinTree_complex ).
thf(sy_c_JoinTree_OjoinTree_OJoin_001t__Nat__Onat,type,
join_nat: joinTree_nat > joinTree_nat > joinTree_nat ).
thf(sy_c_JoinTree_OjoinTree_OJoin_001t__Real__Oreal,type,
join_real: joinTree_real > joinTree_real > joinTree_real ).
thf(sy_c_JoinTree_OjoinTree_OJoin_001tf__a,type,
join_a: joinTree_a > joinTree_a > joinTree_a ).
thf(sy_c_JoinTree_OjoinTree_ORelation_001t__Complex__Ocomplex,type,
relation_complex: complex > joinTree_complex ).
thf(sy_c_JoinTree_OjoinTree_ORelation_001t__Nat__Onat,type,
relation_nat: nat > joinTree_nat ).
thf(sy_c_JoinTree_OjoinTree_ORelation_001t__Real__Oreal,type,
relation_real: real > joinTree_real ).
thf(sy_c_JoinTree_OjoinTree_ORelation_001tf__a,type,
relation_a: a > joinTree_a ).
thf(sy_c_JoinTree_OjoinTree_Orelations_001t__Complex__Ocomplex,type,
relations_complex: joinTree_complex > set_complex ).
thf(sy_c_JoinTree_OjoinTree_Orelations_001t__Nat__Onat,type,
relations_nat: joinTree_nat > set_nat ).
thf(sy_c_JoinTree_OjoinTree_Orelations_001t__Real__Oreal,type,
relations_real: joinTree_real > set_real ).
thf(sy_c_JoinTree_OjoinTree_Orelations_001tf__a,type,
relations_a: joinTree_a > set_a ).
thf(sy_c_JoinTree_OjoinTree_Osize__joinTree_001tf__a,type,
size_joinTree_a: ( a > nat ) > joinTree_a > nat ).
thf(sy_c_JoinTree_Oldeep__T_001tf__a,type,
ldeep_T_a: ( a > real ) > ( a > real ) > list_a > real ).
thf(sy_c_JoinTree_Oldeep__T_H_001tf__a,type,
ldeep_T_a2: ( a > real ) > ( a > real ) > list_a > real ).
thf(sy_c_JoinTree_Oldeep__n_001t__Complex__Ocomplex,type,
ldeep_n_complex: ( complex > complex > real ) > ( complex > real ) > list_complex > real ).
thf(sy_c_JoinTree_Oldeep__n_001t__Nat__Onat,type,
ldeep_n_nat: ( nat > nat > real ) > ( nat > real ) > list_nat > real ).
thf(sy_c_JoinTree_Oldeep__n_001tf__a,type,
ldeep_n_a: ( a > a > real ) > ( a > real ) > list_a > real ).
thf(sy_c_JoinTree_Oleft__deep_001t__Complex__Ocomplex,type,
left_deep_complex: joinTree_complex > $o ).
thf(sy_c_JoinTree_Oleft__deep_001t__Nat__Onat,type,
left_deep_nat: joinTree_nat > $o ).
thf(sy_c_JoinTree_Oleft__deep_001tf__a,type,
left_deep_a: joinTree_a > $o ).
thf(sy_c_JoinTree_Onum__relations_001tf__a,type,
num_relations_a: joinTree_a > nat ).
thf(sy_c_JoinTree_Opos__list__cards_001tf__a,type,
pos_list_cards_a: ( a > real ) > list_a > $o ).
thf(sy_c_JoinTree_Oreasonable__cards_001t__Complex__Ocomplex,type,
reason8668276539946136544omplex: ( complex > real ) > ( complex > complex > real ) > joinTree_complex > $o ).
thf(sy_c_JoinTree_Oreasonable__cards_001t__Nat__Onat,type,
reasonable_cards_nat: ( nat > real ) > ( nat > nat > real ) > joinTree_nat > $o ).
thf(sy_c_JoinTree_Oreasonable__cards_001tf__a,type,
reasonable_cards_a: ( a > real ) > ( a > a > real ) > joinTree_a > $o ).
thf(sy_c_JoinTree_Orevorder_001t__Complex__Ocomplex,type,
revorder_complex: joinTree_complex > list_complex ).
thf(sy_c_JoinTree_Orevorder_001t__Nat__Onat,type,
revorder_nat: joinTree_nat > list_nat ).
thf(sy_c_JoinTree_Orevorder_001tf__a,type,
revorder_a: joinTree_a > list_a ).
thf(sy_c_JoinTree_Oright__deep_001tf__a,type,
right_deep_a: joinTree_a > $o ).
thf(sy_c_List_Oappend_001t__Real__Oreal,type,
append_real: list_real > list_real > list_real ).
thf(sy_c_List_Oappend_001tf__a,type,
append_a: list_a > list_a > list_a ).
thf(sy_c_List_Odistinct_001t__Complex__Ocomplex,type,
distinct_complex: list_complex > $o ).
thf(sy_c_List_Odistinct_001t__Nat__Onat,type,
distinct_nat: list_nat > $o ).
thf(sy_c_List_Odistinct_001tf__a,type,
distinct_a: list_a > $o ).
thf(sy_c_List_Ofoldr_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
foldr_2100044862368902405omplex: ( complex > complex > complex ) > list_complex > complex > complex ).
thf(sy_c_List_Ofoldr_001t__Nat__Onat_001t__Nat__Onat,type,
foldr_nat_nat: ( nat > nat > nat ) > list_nat > nat > nat ).
thf(sy_c_List_Ofoldr_001t__Real__Oreal_001t__Real__Oreal,type,
foldr_real_real: ( real > real > real ) > list_real > real > real ).
thf(sy_c_List_Olast_001tf__a,type,
last_a: list_a > a ).
thf(sy_c_List_Olist_OCons_001t__Real__Oreal,type,
cons_real: real > list_real > list_real ).
thf(sy_c_List_Olist_OCons_001tf__a,type,
cons_a: a > list_a > list_a ).
thf(sy_c_List_Olist_ONil_001tf__a,type,
nil_a: list_a ).
thf(sy_c_List_Olist_Oset_001t__Complex__Ocomplex,type,
set_complex2: list_complex > set_complex ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
set_real2: list_real > set_real ).
thf(sy_c_List_Olist_Oset_001tf__a,type,
set_a2: list_a > set_a ).
thf(sy_c_List_Orev_001tf__a,type,
rev_a: list_a > list_a ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__JoinTree__OjoinTree_Itf__a_J,type,
size_size_joinTree_a: joinTree_a > nat ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Complex__Ocomplex,type,
neg_nu8557863876264182079omplex: complex > complex ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
neg_nu8295874005876285629c_real: real > real ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Complex__Ocomplex_J,type,
ord_le211207098394363844omplex: set_complex > set_complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex,type,
power_power_complex: complex > nat > complex ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
power_power_real: real > nat > real ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex,type,
divide1717551699836669952omplex: complex > complex > complex ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
divide_divide_real: real > real > real ).
thf(sy_c_Selectivities_Oldeep__s_001tf__a,type,
ldeep_s_a: ( a > a > real ) > list_a > a > real ).
thf(sy_c_Selectivities_Olist__sel_001tf__a,type,
list_sel_a: ( a > a > real ) > list_a > list_a > real ).
thf(sy_c_Selectivities_Olist__sel__aux_H_001tf__a,type,
list_sel_aux_a: ( a > a > real ) > list_a > a > real ).
thf(sy_c_Selectivities_Osel__reasonable_001t__Complex__Ocomplex,type,
sel_re4140149273989857872omplex: ( complex > complex > real ) > $o ).
thf(sy_c_Selectivities_Osel__reasonable_001t__Nat__Onat,type,
sel_reasonable_nat: ( nat > nat > real ) > $o ).
thf(sy_c_Selectivities_Osel__reasonable_001tf__a,type,
sel_reasonable_a: ( a > a > real ) > $o ).
thf(sy_c_Selectivities_Osel__symm_001t__Complex__Ocomplex,type,
sel_symm_complex: ( complex > complex > real ) > $o ).
thf(sy_c_Selectivities_Osel__symm_001t__Nat__Onat,type,
sel_symm_nat: ( nat > nat > real ) > $o ).
thf(sy_c_Selectivities_Osel__symm_001tf__a,type,
sel_symm_a: ( a > a > real ) > $o ).
thf(sy_c_Selectivities_Oset__sel_001t__Complex__Ocomplex,type,
set_sel_complex: ( complex > complex > real ) > set_complex > set_complex > real ).
thf(sy_c_Selectivities_Oset__sel_001t__Nat__Onat,type,
set_sel_nat: ( nat > nat > real ) > set_nat > set_nat > real ).
thf(sy_c_Selectivities_Oset__sel_H_001t__Complex__Ocomplex,type,
set_sel_complex2: ( complex > complex > real ) > set_complex > set_complex > real ).
thf(sy_c_Selectivities_Oset__sel_H_001t__Nat__Onat,type,
set_sel_nat2: ( nat > nat > real ) > set_nat > set_nat > real ).
thf(sy_c_Selectivities_Oset__sel__aux_001t__Complex__Ocomplex,type,
set_sel_aux_complex: ( complex > complex > real ) > complex > set_complex > real ).
thf(sy_c_Selectivities_Oset__sel__aux_001t__Nat__Onat,type,
set_sel_aux_nat: ( nat > nat > real ) > nat > set_nat > real ).
thf(sy_c_Selectivities_Oset__sel__aux_H_001t__Complex__Ocomplex,type,
set_sel_aux_complex2: ( complex > complex > real ) > set_complex > complex > real ).
thf(sy_c_Selectivities_Oset__sel__aux_H_001t__Nat__Onat,type,
set_sel_aux_nat2: ( nat > nat > real ) > set_nat > nat > real ).
thf(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
collect_complex: ( complex > $o ) > set_complex ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Complex__Ocomplex_J,type,
collect_set_complex: ( set_complex > $o ) > set_set_complex ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
set_ord_atMost_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Real__Oreal,type,
set_ord_atMost_real: real > set_real ).
thf(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
arcosh_real: real > real ).
thf(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
arsinh_real: real > real ).
thf(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
artanh_real: real > real ).
thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
ln_ln_real: real > real ).
thf(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
powr_real: real > real > real ).
thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_Itf__a_J,type,
accp_list_a: ( list_a > list_a > $o ) > list_a > $o ).
thf(sy_c_member_001t__Complex__Ocomplex,type,
member_complex: complex > set_complex > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__Set__Oset_It__Complex__Ocomplex_J,type,
member_set_complex: set_complex > set_set_complex > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_cf,type,
cf: a > real ).
thf(sy_v_f,type,
f: a > a > real ).
thf(sy_v_h1,type,
h1: a > real > real ).
thf(sy_v_h2,type,
h2: a > real ).
thf(sy_v_l____,type,
l: joinTree_a ).
thf(sy_v_r____,type,
r: a ).
thf(sy_v_t,type,
t: joinTree_a ).
thf(sy_v_xs,type,
xs: list_a ).
% Relevant facts (1264)
thf(fact_0_assms_I2_J,axiom,
distinct_relations_a @ t ).
% assms(2)
thf(fact_1_assms_I4_J,axiom,
left_deep_a @ t ).
% assms(4)
thf(fact_2__092_060open_062distinct__relations_Al_092_060close_062,axiom,
distinct_relations_a @ l ).
% \<open>distinct_relations l\<close>
thf(fact_3__C2_Oprems_C_I3_J,axiom,
left_deep_a @ ( join_a @ l @ ( relation_a @ r ) ) ).
% "2.prems"(3)
thf(fact_4_h1__h2__l,axiom,
! [X: a] :
( ( member_a @ X @ ( relations_a @ l ) )
=> ( ( h1 @ X @ ( cf @ X ) )
= ( h2 @ X ) ) ) ).
% h1_h2_l
thf(fact_5_xs__def,axiom,
( xs
= ( revorder_a @ t ) ) ).
% xs_def
thf(fact_6__C2_Oprems_C_I1_J,axiom,
distinct_relations_a @ ( join_a @ l @ ( relation_a @ r ) ) ).
% "2.prems"(1)
thf(fact_7__C2_Oprems_C_I2_J,axiom,
reasonable_cards_a @ cf @ f @ ( join_a @ l @ ( relation_a @ r ) ) ).
% "2.prems"(2)
thf(fact_8__092_060open_062c__IKKBZ_Ah1_Acf_Af_A_IJoin_Al_A_IRelation_Ar_J_J_A_061_AJoinTree_Ocard_Acf_Af_Al_A_K_Ah1_Ar_A_Icf_Ar_J_A_L_Ac__IKKBZ_Ah1_Acf_Af_Al_092_060close_062,axiom,
( ( c_IKKBZ_a @ h1 @ cf @ f @ ( join_a @ l @ ( relation_a @ r ) ) )
= ( plus_plus_real @ ( times_times_real @ ( card_a @ cf @ f @ l ) @ ( h1 @ r @ ( cf @ r ) ) ) @ ( c_IKKBZ_a @ h1 @ cf @ f @ l ) ) ) ).
% \<open>c_IKKBZ h1 cf f (Join l (Relation r)) = JoinTree.card cf f l * h1 r (cf r) + c_IKKBZ h1 cf f l\<close>
thf(fact_9_assms_I3_J,axiom,
reasonable_cards_a @ cf @ f @ t ).
% assms(3)
thf(fact_10__C2_Oprems_C_I4_J,axiom,
! [X: a] :
( ( member_a @ X @ ( relations_a @ ( join_a @ l @ ( relation_a @ r ) ) ) )
=> ( ( h1 @ X @ ( cf @ X ) )
= ( h2 @ X ) ) ) ).
% "2.prems"(4)
thf(fact_11_assms_I5_J,axiom,
! [X: a] :
( ( member_a @ X @ ( relations_a @ t ) )
=> ( ( h1 @ X @ ( cf @ X ) )
= ( h2 @ X ) ) ) ).
% assms(5)
thf(fact_12__C2_Ohyps_C,axiom,
( ( distinct_relations_a @ l )
=> ( ( reasonable_cards_a @ cf @ f @ l )
=> ( ( left_deep_a @ l )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( relations_a @ l ) )
=> ( ( h1 @ X2 @ ( cf @ X2 ) )
= ( h2 @ X2 ) ) )
=> ( ( c_IKKBZ_a @ h1 @ cf @ f @ l )
= ( c_list_a @ ( ldeep_s_a @ f @ ( revorder_a @ l ) ) @ cf @ h2 @ ( first_node_a @ l ) @ ( revorder_a @ l ) ) ) ) ) ) ) ).
% "2.hyps"
thf(fact_13_c__IKKBZ_Osimps_I2_J,axiom,
! [H: complex > real > real,Cf: complex > real,F: complex > complex > real,L: joinTree_complex,Rel: complex] :
( ( c_IKKBZ_complex @ H @ Cf @ F @ ( join_complex @ L @ ( relation_complex @ Rel ) ) )
= ( plus_plus_real @ ( times_times_real @ ( card_complex @ Cf @ F @ L ) @ ( H @ Rel @ ( Cf @ Rel ) ) ) @ ( c_IKKBZ_complex @ H @ Cf @ F @ L ) ) ) ).
% c_IKKBZ.simps(2)
thf(fact_14_c__IKKBZ_Osimps_I2_J,axiom,
! [H: nat > real > real,Cf: nat > real,F: nat > nat > real,L: joinTree_nat,Rel: nat] :
( ( c_IKKBZ_nat @ H @ Cf @ F @ ( join_nat @ L @ ( relation_nat @ Rel ) ) )
= ( plus_plus_real @ ( times_times_real @ ( card_nat @ Cf @ F @ L ) @ ( H @ Rel @ ( Cf @ Rel ) ) ) @ ( c_IKKBZ_nat @ H @ Cf @ F @ L ) ) ) ).
% c_IKKBZ.simps(2)
thf(fact_15_c__IKKBZ_Osimps_I2_J,axiom,
! [H: a > real > real,Cf: a > real,F: a > a > real,L: joinTree_a,Rel: a] :
( ( c_IKKBZ_a @ H @ Cf @ F @ ( join_a @ L @ ( relation_a @ Rel ) ) )
= ( plus_plus_real @ ( times_times_real @ ( card_a @ Cf @ F @ L ) @ ( H @ Rel @ ( Cf @ Rel ) ) ) @ ( c_IKKBZ_a @ H @ Cf @ F @ L ) ) ) ).
% c_IKKBZ.simps(2)
thf(fact_16_first__node_Oelims,axiom,
! [X3: joinTree_complex,Y: complex] :
( ( ( first_node_complex @ X3 )
= Y )
=> ( ! [R: complex] :
( ( X3
= ( relation_complex @ R ) )
=> ( Y != R ) )
=> ~ ! [L2: joinTree_complex] :
( ? [Uu: joinTree_complex] :
( X3
= ( join_complex @ L2 @ Uu ) )
=> ( Y
!= ( first_node_complex @ L2 ) ) ) ) ) ).
% first_node.elims
thf(fact_17_first__node_Oelims,axiom,
! [X3: joinTree_nat,Y: nat] :
( ( ( first_node_nat @ X3 )
= Y )
=> ( ! [R: nat] :
( ( X3
= ( relation_nat @ R ) )
=> ( Y != R ) )
=> ~ ! [L2: joinTree_nat] :
( ? [Uu: joinTree_nat] :
( X3
= ( join_nat @ L2 @ Uu ) )
=> ( Y
!= ( first_node_nat @ L2 ) ) ) ) ) ).
% first_node.elims
thf(fact_18_first__node_Oelims,axiom,
! [X3: joinTree_a,Y: a] :
( ( ( first_node_a @ X3 )
= Y )
=> ( ! [R: a] :
( ( X3
= ( relation_a @ R ) )
=> ( Y != R ) )
=> ~ ! [L2: joinTree_a] :
( ? [Uu: joinTree_a] :
( X3
= ( join_a @ L2 @ Uu ) )
=> ( Y
!= ( first_node_a @ L2 ) ) ) ) ) ).
% first_node.elims
thf(fact_19_joinTree_Oinject_I1_J,axiom,
! [X1: a,Y1: a] :
( ( ( relation_a @ X1 )
= ( relation_a @ Y1 ) )
= ( X1 = Y1 ) ) ).
% joinTree.inject(1)
thf(fact_20_joinTree_Oinject_I2_J,axiom,
! [X21: joinTree_complex,X22: joinTree_complex,Y21: joinTree_complex,Y22: joinTree_complex] :
( ( ( join_complex @ X21 @ X22 )
= ( join_complex @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% joinTree.inject(2)
thf(fact_21_joinTree_Oinject_I2_J,axiom,
! [X21: joinTree_nat,X22: joinTree_nat,Y21: joinTree_nat,Y22: joinTree_nat] :
( ( ( join_nat @ X21 @ X22 )
= ( join_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% joinTree.inject(2)
thf(fact_22_joinTree_Oinject_I2_J,axiom,
! [X21: joinTree_a,X22: joinTree_a,Y21: joinTree_a,Y22: joinTree_a] :
( ( ( join_a @ X21 @ X22 )
= ( join_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% joinTree.inject(2)
thf(fact_23_first__node_Osimps_I1_J,axiom,
! [R2: a] :
( ( first_node_a @ ( relation_a @ R2 ) )
= R2 ) ).
% first_node.simps(1)
thf(fact_24_first__node_Osimps_I2_J,axiom,
! [L: joinTree_complex,Uu2: joinTree_complex] :
( ( first_node_complex @ ( join_complex @ L @ Uu2 ) )
= ( first_node_complex @ L ) ) ).
% first_node.simps(2)
thf(fact_25_first__node_Osimps_I2_J,axiom,
! [L: joinTree_nat,Uu2: joinTree_nat] :
( ( first_node_nat @ ( join_nat @ L @ Uu2 ) )
= ( first_node_nat @ L ) ) ).
% first_node.simps(2)
thf(fact_26_first__node_Osimps_I2_J,axiom,
! [L: joinTree_a,Uu2: joinTree_a] :
( ( first_node_a @ ( join_a @ L @ Uu2 ) )
= ( first_node_a @ L ) ) ).
% first_node.simps(2)
thf(fact_27_add__left__cancel,axiom,
! [A: complex,B: complex,C: complex] :
( ( ( plus_plus_complex @ A @ B )
= ( plus_plus_complex @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_28_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_29_add__left__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_30_add__right__cancel,axiom,
! [B: complex,A: complex,C: complex] :
( ( ( plus_plus_complex @ B @ A )
= ( plus_plus_complex @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_31_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_32_add__right__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_33_card_Osimps_I1_J,axiom,
! [Cf: a > real,F: a > a > real,Rel: a] :
( ( card_a @ Cf @ F @ ( relation_a @ Rel ) )
= ( Cf @ Rel ) ) ).
% card.simps(1)
thf(fact_34_joinTree_Odistinct_I1_J,axiom,
! [X1: complex,X21: joinTree_complex,X22: joinTree_complex] :
( ( relation_complex @ X1 )
!= ( join_complex @ X21 @ X22 ) ) ).
% joinTree.distinct(1)
thf(fact_35_joinTree_Odistinct_I1_J,axiom,
! [X1: nat,X21: joinTree_nat,X22: joinTree_nat] :
( ( relation_nat @ X1 )
!= ( join_nat @ X21 @ X22 ) ) ).
% joinTree.distinct(1)
thf(fact_36_joinTree_Odistinct_I1_J,axiom,
! [X1: a,X21: joinTree_a,X22: joinTree_a] :
( ( relation_a @ X1 )
!= ( join_a @ X21 @ X22 ) ) ).
% joinTree.distinct(1)
thf(fact_37_zig__zag_Ocases,axiom,
! [X3: joinTree_complex] :
( ! [Uu: complex] :
( X3
!= ( relation_complex @ Uu ) )
=> ( ! [L2: joinTree_complex,Uv: complex] :
( X3
!= ( join_complex @ L2 @ ( relation_complex @ Uv ) ) )
=> ( ! [Uw: complex,V: joinTree_complex,Va: joinTree_complex] :
( X3
!= ( join_complex @ ( relation_complex @ Uw ) @ ( join_complex @ V @ Va ) ) )
=> ~ ! [Va: joinTree_complex,Vd: joinTree_complex,Vb: joinTree_complex,Vc: joinTree_complex] :
( X3
!= ( join_complex @ ( join_complex @ Va @ Vd ) @ ( join_complex @ Vb @ Vc ) ) ) ) ) ) ).
% zig_zag.cases
thf(fact_38_zig__zag_Ocases,axiom,
! [X3: joinTree_nat] :
( ! [Uu: nat] :
( X3
!= ( relation_nat @ Uu ) )
=> ( ! [L2: joinTree_nat,Uv: nat] :
( X3
!= ( join_nat @ L2 @ ( relation_nat @ Uv ) ) )
=> ( ! [Uw: nat,V: joinTree_nat,Va: joinTree_nat] :
( X3
!= ( join_nat @ ( relation_nat @ Uw ) @ ( join_nat @ V @ Va ) ) )
=> ~ ! [Va: joinTree_nat,Vd: joinTree_nat,Vb: joinTree_nat,Vc: joinTree_nat] :
( X3
!= ( join_nat @ ( join_nat @ Va @ Vd ) @ ( join_nat @ Vb @ Vc ) ) ) ) ) ) ).
% zig_zag.cases
thf(fact_39_zig__zag_Ocases,axiom,
! [X3: joinTree_a] :
( ! [Uu: a] :
( X3
!= ( relation_a @ Uu ) )
=> ( ! [L2: joinTree_a,Uv: a] :
( X3
!= ( join_a @ L2 @ ( relation_a @ Uv ) ) )
=> ( ! [Uw: a,V: joinTree_a,Va: joinTree_a] :
( X3
!= ( join_a @ ( relation_a @ Uw ) @ ( join_a @ V @ Va ) ) )
=> ~ ! [Va: joinTree_a,Vd: joinTree_a,Vb: joinTree_a,Vc: joinTree_a] :
( X3
!= ( join_a @ ( join_a @ Va @ Vd ) @ ( join_a @ Vb @ Vc ) ) ) ) ) ) ).
% zig_zag.cases
thf(fact_40_distinct__elem__right__not__left,axiom,
! [L: joinTree_real,R2: joinTree_real,X3: real] :
( ( distin8564637588129275604s_real @ ( join_real @ L @ R2 ) )
=> ( ( member_real @ X3 @ ( relations_real @ R2 ) )
=> ~ ( member_real @ X3 @ ( relations_real @ L ) ) ) ) ).
% distinct_elem_right_not_left
thf(fact_41_distinct__elem__right__not__left,axiom,
! [L: joinTree_complex,R2: joinTree_complex,X3: complex] :
( ( distin1760132951325961430omplex @ ( join_complex @ L @ R2 ) )
=> ( ( member_complex @ X3 @ ( relations_complex @ R2 ) )
=> ~ ( member_complex @ X3 @ ( relations_complex @ L ) ) ) ) ).
% distinct_elem_right_not_left
thf(fact_42_distinct__elem__right__not__left,axiom,
! [L: joinTree_nat,R2: joinTree_nat,X3: nat] :
( ( distin9127876880396718072ns_nat @ ( join_nat @ L @ R2 ) )
=> ( ( member_nat @ X3 @ ( relations_nat @ R2 ) )
=> ~ ( member_nat @ X3 @ ( relations_nat @ L ) ) ) ) ).
% distinct_elem_right_not_left
thf(fact_43_distinct__elem__right__not__left,axiom,
! [L: joinTree_a,R2: joinTree_a,X3: a] :
( ( distinct_relations_a @ ( join_a @ L @ R2 ) )
=> ( ( member_a @ X3 @ ( relations_a @ R2 ) )
=> ~ ( member_a @ X3 @ ( relations_a @ L ) ) ) ) ).
% distinct_elem_right_not_left
thf(fact_44_distinct__elem__left__not__right,axiom,
! [L: joinTree_real,R2: joinTree_real,X3: real] :
( ( distin8564637588129275604s_real @ ( join_real @ L @ R2 ) )
=> ( ( member_real @ X3 @ ( relations_real @ L ) )
=> ~ ( member_real @ X3 @ ( relations_real @ R2 ) ) ) ) ).
% distinct_elem_left_not_right
thf(fact_45_distinct__elem__left__not__right,axiom,
! [L: joinTree_complex,R2: joinTree_complex,X3: complex] :
( ( distin1760132951325961430omplex @ ( join_complex @ L @ R2 ) )
=> ( ( member_complex @ X3 @ ( relations_complex @ L ) )
=> ~ ( member_complex @ X3 @ ( relations_complex @ R2 ) ) ) ) ).
% distinct_elem_left_not_right
thf(fact_46_distinct__elem__left__not__right,axiom,
! [L: joinTree_nat,R2: joinTree_nat,X3: nat] :
( ( distin9127876880396718072ns_nat @ ( join_nat @ L @ R2 ) )
=> ( ( member_nat @ X3 @ ( relations_nat @ L ) )
=> ~ ( member_nat @ X3 @ ( relations_nat @ R2 ) ) ) ) ).
% distinct_elem_left_not_right
thf(fact_47_distinct__elem__left__not__right,axiom,
! [L: joinTree_a,R2: joinTree_a,X3: a] :
( ( distinct_relations_a @ ( join_a @ L @ R2 ) )
=> ( ( member_a @ X3 @ ( relations_a @ L ) )
=> ~ ( member_a @ X3 @ ( relations_a @ R2 ) ) ) ) ).
% distinct_elem_left_not_right
thf(fact_48_joinTree_Oset__intros_I2_J,axiom,
! [Y: real,X21: joinTree_real,X22: joinTree_real] :
( ( member_real @ Y @ ( relations_real @ X21 ) )
=> ( member_real @ Y @ ( relations_real @ ( join_real @ X21 @ X22 ) ) ) ) ).
% joinTree.set_intros(2)
thf(fact_49_joinTree_Oset__intros_I2_J,axiom,
! [Y: complex,X21: joinTree_complex,X22: joinTree_complex] :
( ( member_complex @ Y @ ( relations_complex @ X21 ) )
=> ( member_complex @ Y @ ( relations_complex @ ( join_complex @ X21 @ X22 ) ) ) ) ).
% joinTree.set_intros(2)
thf(fact_50_joinTree_Oset__intros_I2_J,axiom,
! [Y: nat,X21: joinTree_nat,X22: joinTree_nat] :
( ( member_nat @ Y @ ( relations_nat @ X21 ) )
=> ( member_nat @ Y @ ( relations_nat @ ( join_nat @ X21 @ X22 ) ) ) ) ).
% joinTree.set_intros(2)
thf(fact_51_joinTree_Oset__intros_I2_J,axiom,
! [Y: a,X21: joinTree_a,X22: joinTree_a] :
( ( member_a @ Y @ ( relations_a @ X21 ) )
=> ( member_a @ Y @ ( relations_a @ ( join_a @ X21 @ X22 ) ) ) ) ).
% joinTree.set_intros(2)
thf(fact_52_joinTree_Oset__intros_I3_J,axiom,
! [Ya: real,X22: joinTree_real,X21: joinTree_real] :
( ( member_real @ Ya @ ( relations_real @ X22 ) )
=> ( member_real @ Ya @ ( relations_real @ ( join_real @ X21 @ X22 ) ) ) ) ).
% joinTree.set_intros(3)
thf(fact_53_joinTree_Oset__intros_I3_J,axiom,
! [Ya: complex,X22: joinTree_complex,X21: joinTree_complex] :
( ( member_complex @ Ya @ ( relations_complex @ X22 ) )
=> ( member_complex @ Ya @ ( relations_complex @ ( join_complex @ X21 @ X22 ) ) ) ) ).
% joinTree.set_intros(3)
thf(fact_54_joinTree_Oset__intros_I3_J,axiom,
! [Ya: nat,X22: joinTree_nat,X21: joinTree_nat] :
( ( member_nat @ Ya @ ( relations_nat @ X22 ) )
=> ( member_nat @ Ya @ ( relations_nat @ ( join_nat @ X21 @ X22 ) ) ) ) ).
% joinTree.set_intros(3)
thf(fact_55_joinTree_Oset__intros_I3_J,axiom,
! [Ya: a,X22: joinTree_a,X21: joinTree_a] :
( ( member_a @ Ya @ ( relations_a @ X22 ) )
=> ( member_a @ Ya @ ( relations_a @ ( join_a @ X21 @ X22 ) ) ) ) ).
% joinTree.set_intros(3)
thf(fact_56_joinTree_Oset__intros_I1_J,axiom,
! [X1: real] : ( member_real @ X1 @ ( relations_real @ ( relation_real @ X1 ) ) ) ).
% joinTree.set_intros(1)
thf(fact_57_joinTree_Oset__intros_I1_J,axiom,
! [X1: complex] : ( member_complex @ X1 @ ( relations_complex @ ( relation_complex @ X1 ) ) ) ).
% joinTree.set_intros(1)
thf(fact_58_joinTree_Oset__intros_I1_J,axiom,
! [X1: nat] : ( member_nat @ X1 @ ( relations_nat @ ( relation_nat @ X1 ) ) ) ).
% joinTree.set_intros(1)
thf(fact_59_joinTree_Oset__intros_I1_J,axiom,
! [X1: a] : ( member_a @ X1 @ ( relations_a @ ( relation_a @ X1 ) ) ) ).
% joinTree.set_intros(1)
thf(fact_60_reasonable__trans,axiom,
! [Cf: complex > real,F: complex > complex > real,L: joinTree_complex,R2: joinTree_complex] :
( ( reason8668276539946136544omplex @ Cf @ F @ ( join_complex @ L @ R2 ) )
=> ( ( reason8668276539946136544omplex @ Cf @ F @ L )
& ( reason8668276539946136544omplex @ Cf @ F @ R2 ) ) ) ).
% reasonable_trans
thf(fact_61_reasonable__trans,axiom,
! [Cf: nat > real,F: nat > nat > real,L: joinTree_nat,R2: joinTree_nat] :
( ( reasonable_cards_nat @ Cf @ F @ ( join_nat @ L @ R2 ) )
=> ( ( reasonable_cards_nat @ Cf @ F @ L )
& ( reasonable_cards_nat @ Cf @ F @ R2 ) ) ) ).
% reasonable_trans
thf(fact_62_reasonable__trans,axiom,
! [Cf: a > real,F: a > a > real,L: joinTree_a,R2: joinTree_a] :
( ( reasonable_cards_a @ Cf @ F @ ( join_a @ L @ R2 ) )
=> ( ( reasonable_cards_a @ Cf @ F @ L )
& ( reasonable_cards_a @ Cf @ F @ R2 ) ) ) ).
% reasonable_trans
thf(fact_63_ldeep__trans,axiom,
! [L: joinTree_complex,R2: joinTree_complex] :
( ( left_deep_complex @ ( join_complex @ L @ R2 ) )
=> ( left_deep_complex @ L ) ) ).
% ldeep_trans
thf(fact_64_ldeep__trans,axiom,
! [L: joinTree_nat,R2: joinTree_nat] :
( ( left_deep_nat @ ( join_nat @ L @ R2 ) )
=> ( left_deep_nat @ L ) ) ).
% ldeep_trans
thf(fact_65_ldeep__trans,axiom,
! [L: joinTree_a,R2: joinTree_a] :
( ( left_deep_a @ ( join_a @ L @ R2 ) )
=> ( left_deep_a @ L ) ) ).
% ldeep_trans
thf(fact_66_left__deep_Osimps_I3_J,axiom,
! [V2: joinTree_complex,Vb2: joinTree_complex,Vc2: joinTree_complex] :
~ ( left_deep_complex @ ( join_complex @ V2 @ ( join_complex @ Vb2 @ Vc2 ) ) ) ).
% left_deep.simps(3)
thf(fact_67_left__deep_Osimps_I3_J,axiom,
! [V2: joinTree_nat,Vb2: joinTree_nat,Vc2: joinTree_nat] :
~ ( left_deep_nat @ ( join_nat @ V2 @ ( join_nat @ Vb2 @ Vc2 ) ) ) ).
% left_deep.simps(3)
thf(fact_68_left__deep_Osimps_I3_J,axiom,
! [V2: joinTree_a,Vb2: joinTree_a,Vc2: joinTree_a] :
~ ( left_deep_a @ ( join_a @ V2 @ ( join_a @ Vb2 @ Vc2 ) ) ) ).
% left_deep.simps(3)
thf(fact_69_distinct__trans__r,axiom,
! [L: joinTree_complex,R2: joinTree_complex] :
( ( distin1760132951325961430omplex @ ( join_complex @ L @ R2 ) )
=> ( distin1760132951325961430omplex @ R2 ) ) ).
% distinct_trans_r
thf(fact_70_distinct__trans__r,axiom,
! [L: joinTree_nat,R2: joinTree_nat] :
( ( distin9127876880396718072ns_nat @ ( join_nat @ L @ R2 ) )
=> ( distin9127876880396718072ns_nat @ R2 ) ) ).
% distinct_trans_r
thf(fact_71_distinct__trans__r,axiom,
! [L: joinTree_a,R2: joinTree_a] :
( ( distinct_relations_a @ ( join_a @ L @ R2 ) )
=> ( distinct_relations_a @ R2 ) ) ).
% distinct_trans_r
thf(fact_72_distinct__trans__l,axiom,
! [L: joinTree_complex,R2: joinTree_complex] :
( ( distin1760132951325961430omplex @ ( join_complex @ L @ R2 ) )
=> ( distin1760132951325961430omplex @ L ) ) ).
% distinct_trans_l
thf(fact_73_distinct__trans__l,axiom,
! [L: joinTree_nat,R2: joinTree_nat] :
( ( distin9127876880396718072ns_nat @ ( join_nat @ L @ R2 ) )
=> ( distin9127876880396718072ns_nat @ L ) ) ).
% distinct_trans_l
thf(fact_74_distinct__trans__l,axiom,
! [L: joinTree_a,R2: joinTree_a] :
( ( distinct_relations_a @ ( join_a @ L @ R2 ) )
=> ( distinct_relations_a @ L ) ) ).
% distinct_trans_l
thf(fact_75_left__deep_Osimps_I1_J,axiom,
! [Uu2: a] : ( left_deep_a @ ( relation_a @ Uu2 ) ) ).
% left_deep.simps(1)
thf(fact_76_mult_Oleft__commute,axiom,
! [B: complex,A: complex,C: complex] :
( ( times_times_complex @ B @ ( times_times_complex @ A @ C ) )
= ( times_times_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_77_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_78_mult_Oleft__commute,axiom,
! [B: real,A: real,C: real] :
( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_79_mult_Ocommute,axiom,
( times_times_complex
= ( ^ [A2: complex,B2: complex] : ( times_times_complex @ B2 @ A2 ) ) ) ).
% mult.commute
thf(fact_80_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A2: nat,B2: nat] : ( times_times_nat @ B2 @ A2 ) ) ) ).
% mult.commute
thf(fact_81_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A2: real,B2: real] : ( times_times_real @ B2 @ A2 ) ) ) ).
% mult.commute
thf(fact_82_mult_Oassoc,axiom,
! [A: complex,B: complex,C: complex] :
( ( times_times_complex @ ( times_times_complex @ A @ B ) @ C )
= ( times_times_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% mult.assoc
thf(fact_83_mult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_84_mult_Oassoc,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% mult.assoc
thf(fact_85_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: complex,B: complex,C: complex] :
( ( times_times_complex @ ( times_times_complex @ A @ B ) @ C )
= ( times_times_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_86_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_87_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_88_add__right__imp__eq,axiom,
! [B: complex,A: complex,C: complex] :
( ( ( plus_plus_complex @ B @ A )
= ( plus_plus_complex @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_89_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_90_add__right__imp__eq,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_91_add__left__imp__eq,axiom,
! [A: complex,B: complex,C: complex] :
( ( ( plus_plus_complex @ A @ B )
= ( plus_plus_complex @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_92_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_93_add__left__imp__eq,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_94_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_95_mem__Collect__eq,axiom,
! [A: set_complex,P: set_complex > $o] :
( ( member_set_complex @ A @ ( collect_set_complex @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_96_mem__Collect__eq,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( member_set_nat @ A @ ( collect_set_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_97_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_98_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_99_mem__Collect__eq,axiom,
! [A: complex,P: complex > $o] :
( ( member_complex @ A @ ( collect_complex @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_100_Collect__mem__eq,axiom,
! [A3: set_a] :
( ( collect_a
@ ^ [X4: a] : ( member_a @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_101_Collect__mem__eq,axiom,
! [A3: set_set_complex] :
( ( collect_set_complex
@ ^ [X4: set_complex] : ( member_set_complex @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_102_Collect__mem__eq,axiom,
! [A3: set_set_nat] :
( ( collect_set_nat
@ ^ [X4: set_nat] : ( member_set_nat @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_103_Collect__mem__eq,axiom,
! [A3: set_real] :
( ( collect_real
@ ^ [X4: real] : ( member_real @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_104_Collect__mem__eq,axiom,
! [A3: set_nat] :
( ( collect_nat
@ ^ [X4: nat] : ( member_nat @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_105_Collect__mem__eq,axiom,
! [A3: set_complex] :
( ( collect_complex
@ ^ [X4: complex] : ( member_complex @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_106_Collect__cong,axiom,
! [P: set_complex > $o,Q: set_complex > $o] :
( ! [X2: set_complex] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_set_complex @ P )
= ( collect_set_complex @ Q ) ) ) ).
% Collect_cong
thf(fact_107_Collect__cong,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ! [X2: set_nat] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_set_nat @ P )
= ( collect_set_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_108_Collect__cong,axiom,
! [P: real > $o,Q: real > $o] :
( ! [X2: real] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_real @ P )
= ( collect_real @ Q ) ) ) ).
% Collect_cong
thf(fact_109_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_110_Collect__cong,axiom,
! [P: complex > $o,Q: complex > $o] :
( ! [X2: complex] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_complex @ P )
= ( collect_complex @ Q ) ) ) ).
% Collect_cong
thf(fact_111_add_Oleft__commute,axiom,
! [B: complex,A: complex,C: complex] :
( ( plus_plus_complex @ B @ ( plus_plus_complex @ A @ C ) )
= ( plus_plus_complex @ A @ ( plus_plus_complex @ B @ C ) ) ) ).
% add.left_commute
thf(fact_112_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_113_add_Oleft__commute,axiom,
! [B: real,A: real,C: real] :
( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% add.left_commute
thf(fact_114_add_Ocommute,axiom,
( plus_plus_complex
= ( ^ [A2: complex,B2: complex] : ( plus_plus_complex @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_115_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A2: nat,B2: nat] : ( plus_plus_nat @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_116_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A2: real,B2: real] : ( plus_plus_real @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_117_add_Oright__cancel,axiom,
! [B: complex,A: complex,C: complex] :
( ( ( plus_plus_complex @ B @ A )
= ( plus_plus_complex @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_118_add_Oright__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_119_add_Oleft__cancel,axiom,
! [A: complex,B: complex,C: complex] :
( ( ( plus_plus_complex @ A @ B )
= ( plus_plus_complex @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_120_add_Oleft__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_121_add_Oassoc,axiom,
! [A: complex,B: complex,C: complex] :
( ( plus_plus_complex @ ( plus_plus_complex @ A @ B ) @ C )
= ( plus_plus_complex @ A @ ( plus_plus_complex @ B @ C ) ) ) ).
% add.assoc
thf(fact_122_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_123_add_Oassoc,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% add.assoc
thf(fact_124_group__cancel_Oadd2,axiom,
! [B3: complex,K: complex,B: complex,A: complex] :
( ( B3
= ( plus_plus_complex @ K @ B ) )
=> ( ( plus_plus_complex @ A @ B3 )
= ( plus_plus_complex @ K @ ( plus_plus_complex @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_125_group__cancel_Oadd2,axiom,
! [B3: nat,K: nat,B: nat,A: nat] :
( ( B3
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B3 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_126_group__cancel_Oadd2,axiom,
! [B3: real,K: real,B: real,A: real] :
( ( B3
= ( plus_plus_real @ K @ B ) )
=> ( ( plus_plus_real @ A @ B3 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_127_group__cancel_Oadd1,axiom,
! [A3: complex,K: complex,A: complex,B: complex] :
( ( A3
= ( plus_plus_complex @ K @ A ) )
=> ( ( plus_plus_complex @ A3 @ B )
= ( plus_plus_complex @ K @ ( plus_plus_complex @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_128_group__cancel_Oadd1,axiom,
! [A3: nat,K: nat,A: nat,B: nat] :
( ( A3
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A3 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_129_group__cancel_Oadd1,axiom,
! [A3: real,K: real,A: real,B: real] :
( ( A3
= ( plus_plus_real @ K @ A ) )
=> ( ( plus_plus_real @ A3 @ B )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_130_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_131_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_real @ I @ K )
= ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_132_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: complex,B: complex,C: complex] :
( ( plus_plus_complex @ ( plus_plus_complex @ A @ B ) @ C )
= ( plus_plus_complex @ A @ ( plus_plus_complex @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_133_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_134_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_135_joinTree_Oset__cases,axiom,
! [E: real,A: joinTree_real] :
( ( member_real @ E @ ( relations_real @ A ) )
=> ( ( A
!= ( relation_real @ E ) )
=> ( ! [Z1: joinTree_real] :
( ? [Z2: joinTree_real] :
( A
= ( join_real @ Z1 @ Z2 ) )
=> ~ ( member_real @ E @ ( relations_real @ Z1 ) ) )
=> ~ ! [Z1: joinTree_real,Z2: joinTree_real] :
( ( A
= ( join_real @ Z1 @ Z2 ) )
=> ~ ( member_real @ E @ ( relations_real @ Z2 ) ) ) ) ) ) ).
% joinTree.set_cases
thf(fact_136_joinTree_Oset__cases,axiom,
! [E: complex,A: joinTree_complex] :
( ( member_complex @ E @ ( relations_complex @ A ) )
=> ( ( A
!= ( relation_complex @ E ) )
=> ( ! [Z1: joinTree_complex] :
( ? [Z2: joinTree_complex] :
( A
= ( join_complex @ Z1 @ Z2 ) )
=> ~ ( member_complex @ E @ ( relations_complex @ Z1 ) ) )
=> ~ ! [Z1: joinTree_complex,Z2: joinTree_complex] :
( ( A
= ( join_complex @ Z1 @ Z2 ) )
=> ~ ( member_complex @ E @ ( relations_complex @ Z2 ) ) ) ) ) ) ).
% joinTree.set_cases
thf(fact_137_joinTree_Oset__cases,axiom,
! [E: nat,A: joinTree_nat] :
( ( member_nat @ E @ ( relations_nat @ A ) )
=> ( ( A
!= ( relation_nat @ E ) )
=> ( ! [Z1: joinTree_nat] :
( ? [Z2: joinTree_nat] :
( A
= ( join_nat @ Z1 @ Z2 ) )
=> ~ ( member_nat @ E @ ( relations_nat @ Z1 ) ) )
=> ~ ! [Z1: joinTree_nat,Z2: joinTree_nat] :
( ( A
= ( join_nat @ Z1 @ Z2 ) )
=> ~ ( member_nat @ E @ ( relations_nat @ Z2 ) ) ) ) ) ) ).
% joinTree.set_cases
thf(fact_138_joinTree_Oset__cases,axiom,
! [E: a,A: joinTree_a] :
( ( member_a @ E @ ( relations_a @ A ) )
=> ( ( A
!= ( relation_a @ E ) )
=> ( ! [Z1: joinTree_a] :
( ? [Z2: joinTree_a] :
( A
= ( join_a @ Z1 @ Z2 ) )
=> ~ ( member_a @ E @ ( relations_a @ Z1 ) ) )
=> ~ ! [Z1: joinTree_a,Z2: joinTree_a] :
( ( A
= ( join_a @ Z1 @ Z2 ) )
=> ~ ( member_a @ E @ ( relations_a @ Z2 ) ) ) ) ) ) ).
% joinTree.set_cases
thf(fact_139_joinTree__cases__ldeep,axiom,
! [T: joinTree_complex] :
( ( left_deep_complex @ T )
=> ( ? [R: complex] :
( T
= ( relation_complex @ R ) )
| ? [L2: joinTree_complex,Rr: complex] :
( T
= ( join_complex @ L2 @ ( relation_complex @ Rr ) ) ) ) ) ).
% joinTree_cases_ldeep
thf(fact_140_joinTree__cases__ldeep,axiom,
! [T: joinTree_nat] :
( ( left_deep_nat @ T )
=> ( ? [R: nat] :
( T
= ( relation_nat @ R ) )
| ? [L2: joinTree_nat,Rr: nat] :
( T
= ( join_nat @ L2 @ ( relation_nat @ Rr ) ) ) ) ) ).
% joinTree_cases_ldeep
thf(fact_141_joinTree__cases__ldeep,axiom,
! [T: joinTree_a] :
( ( left_deep_a @ T )
=> ( ? [R: a] :
( T
= ( relation_a @ R ) )
| ? [L2: joinTree_a,Rr: a] :
( T
= ( join_a @ L2 @ ( relation_a @ Rr ) ) ) ) ) ).
% joinTree_cases_ldeep
thf(fact_142_left__deep_Osimps_I2_J,axiom,
! [L: joinTree_complex,Uv2: complex] :
( ( left_deep_complex @ ( join_complex @ L @ ( relation_complex @ Uv2 ) ) )
= ( left_deep_complex @ L ) ) ).
% left_deep.simps(2)
thf(fact_143_left__deep_Osimps_I2_J,axiom,
! [L: joinTree_nat,Uv2: nat] :
( ( left_deep_nat @ ( join_nat @ L @ ( relation_nat @ Uv2 ) ) )
= ( left_deep_nat @ L ) ) ).
% left_deep.simps(2)
thf(fact_144_left__deep_Osimps_I2_J,axiom,
! [L: joinTree_a,Uv2: a] :
( ( left_deep_a @ ( join_a @ L @ ( relation_a @ Uv2 ) ) )
= ( left_deep_a @ L ) ) ).
% left_deep.simps(2)
thf(fact_145_left__deep_Oelims_I1_J,axiom,
! [X3: joinTree_complex,Y: $o] :
( ( ( left_deep_complex @ X3 )
= Y )
=> ( ( ? [Uu: complex] :
( X3
= ( relation_complex @ Uu ) )
=> ~ Y )
=> ( ! [L2: joinTree_complex] :
( ? [Uv: complex] :
( X3
= ( join_complex @ L2 @ ( relation_complex @ Uv ) ) )
=> ( Y
= ( ~ ( left_deep_complex @ L2 ) ) ) )
=> ~ ( ? [V: joinTree_complex,Vb: joinTree_complex,Vc: joinTree_complex] :
( X3
= ( join_complex @ V @ ( join_complex @ Vb @ Vc ) ) )
=> Y ) ) ) ) ).
% left_deep.elims(1)
thf(fact_146_left__deep_Oelims_I1_J,axiom,
! [X3: joinTree_nat,Y: $o] :
( ( ( left_deep_nat @ X3 )
= Y )
=> ( ( ? [Uu: nat] :
( X3
= ( relation_nat @ Uu ) )
=> ~ Y )
=> ( ! [L2: joinTree_nat] :
( ? [Uv: nat] :
( X3
= ( join_nat @ L2 @ ( relation_nat @ Uv ) ) )
=> ( Y
= ( ~ ( left_deep_nat @ L2 ) ) ) )
=> ~ ( ? [V: joinTree_nat,Vb: joinTree_nat,Vc: joinTree_nat] :
( X3
= ( join_nat @ V @ ( join_nat @ Vb @ Vc ) ) )
=> Y ) ) ) ) ).
% left_deep.elims(1)
thf(fact_147_left__deep_Oelims_I1_J,axiom,
! [X3: joinTree_a,Y: $o] :
( ( ( left_deep_a @ X3 )
= Y )
=> ( ( ? [Uu: a] :
( X3
= ( relation_a @ Uu ) )
=> ~ Y )
=> ( ! [L2: joinTree_a] :
( ? [Uv: a] :
( X3
= ( join_a @ L2 @ ( relation_a @ Uv ) ) )
=> ( Y
= ( ~ ( left_deep_a @ L2 ) ) ) )
=> ~ ( ? [V: joinTree_a,Vb: joinTree_a,Vc: joinTree_a] :
( X3
= ( join_a @ V @ ( join_a @ Vb @ Vc ) ) )
=> Y ) ) ) ) ).
% left_deep.elims(1)
thf(fact_148_left__deep_Oelims_I2_J,axiom,
! [X3: joinTree_complex] :
( ( left_deep_complex @ X3 )
=> ( ! [Uu: complex] :
( X3
!= ( relation_complex @ Uu ) )
=> ~ ! [L2: joinTree_complex] :
( ? [Uv: complex] :
( X3
= ( join_complex @ L2 @ ( relation_complex @ Uv ) ) )
=> ~ ( left_deep_complex @ L2 ) ) ) ) ).
% left_deep.elims(2)
thf(fact_149_left__deep_Oelims_I2_J,axiom,
! [X3: joinTree_nat] :
( ( left_deep_nat @ X3 )
=> ( ! [Uu: nat] :
( X3
!= ( relation_nat @ Uu ) )
=> ~ ! [L2: joinTree_nat] :
( ? [Uv: nat] :
( X3
= ( join_nat @ L2 @ ( relation_nat @ Uv ) ) )
=> ~ ( left_deep_nat @ L2 ) ) ) ) ).
% left_deep.elims(2)
thf(fact_150_left__deep_Oelims_I2_J,axiom,
! [X3: joinTree_a] :
( ( left_deep_a @ X3 )
=> ( ! [Uu: a] :
( X3
!= ( relation_a @ Uu ) )
=> ~ ! [L2: joinTree_a] :
( ? [Uv: a] :
( X3
= ( join_a @ L2 @ ( relation_a @ Uv ) ) )
=> ~ ( left_deep_a @ L2 ) ) ) ) ).
% left_deep.elims(2)
thf(fact_151_left__deep_Oelims_I3_J,axiom,
! [X3: joinTree_complex] :
( ~ ( left_deep_complex @ X3 )
=> ( ! [L2: joinTree_complex] :
( ? [Uv: complex] :
( X3
= ( join_complex @ L2 @ ( relation_complex @ Uv ) ) )
=> ( left_deep_complex @ L2 ) )
=> ~ ! [V: joinTree_complex,Vb: joinTree_complex,Vc: joinTree_complex] :
( X3
!= ( join_complex @ V @ ( join_complex @ Vb @ Vc ) ) ) ) ) ).
% left_deep.elims(3)
thf(fact_152_left__deep_Oelims_I3_J,axiom,
! [X3: joinTree_nat] :
( ~ ( left_deep_nat @ X3 )
=> ( ! [L2: joinTree_nat] :
( ? [Uv: nat] :
( X3
= ( join_nat @ L2 @ ( relation_nat @ Uv ) ) )
=> ( left_deep_nat @ L2 ) )
=> ~ ! [V: joinTree_nat,Vb: joinTree_nat,Vc: joinTree_nat] :
( X3
!= ( join_nat @ V @ ( join_nat @ Vb @ Vc ) ) ) ) ) ).
% left_deep.elims(3)
thf(fact_153_left__deep_Oelims_I3_J,axiom,
! [X3: joinTree_a] :
( ~ ( left_deep_a @ X3 )
=> ( ! [L2: joinTree_a] :
( ? [Uv: a] :
( X3
= ( join_a @ L2 @ ( relation_a @ Uv ) ) )
=> ( left_deep_a @ L2 ) )
=> ~ ! [V: joinTree_a,Vb: joinTree_a,Vc: joinTree_a] :
( X3
!= ( join_a @ V @ ( join_a @ Vb @ Vc ) ) ) ) ) ).
% left_deep.elims(3)
thf(fact_154_joinTree__cases,axiom,
! [T: joinTree_complex] :
( ? [R: complex] :
( T
= ( relation_complex @ R ) )
| ? [L2: joinTree_complex,Rr: complex] :
( T
= ( join_complex @ L2 @ ( relation_complex @ Rr ) ) )
| ? [L2: joinTree_complex,Lr: joinTree_complex,Rr: joinTree_complex] :
( T
= ( join_complex @ L2 @ ( join_complex @ Lr @ Rr ) ) ) ) ).
% joinTree_cases
thf(fact_155_joinTree__cases,axiom,
! [T: joinTree_nat] :
( ? [R: nat] :
( T
= ( relation_nat @ R ) )
| ? [L2: joinTree_nat,Rr: nat] :
( T
= ( join_nat @ L2 @ ( relation_nat @ Rr ) ) )
| ? [L2: joinTree_nat,Lr: joinTree_nat,Rr: joinTree_nat] :
( T
= ( join_nat @ L2 @ ( join_nat @ Lr @ Rr ) ) ) ) ).
% joinTree_cases
thf(fact_156_joinTree__cases,axiom,
! [T: joinTree_a] :
( ? [R: a] :
( T
= ( relation_a @ R ) )
| ? [L2: joinTree_a,Rr: a] :
( T
= ( join_a @ L2 @ ( relation_a @ Rr ) ) )
| ? [L2: joinTree_a,Lr: joinTree_a,Rr: joinTree_a] :
( T
= ( join_a @ L2 @ ( join_a @ Lr @ Rr ) ) ) ) ).
% joinTree_cases
thf(fact_157_relations__mset_Ocases,axiom,
! [X3: joinTree_complex] :
( ! [Rel2: complex] :
( X3
!= ( relation_complex @ Rel2 ) )
=> ~ ! [L2: joinTree_complex,R: joinTree_complex] :
( X3
!= ( join_complex @ L2 @ R ) ) ) ).
% relations_mset.cases
thf(fact_158_relations__mset_Ocases,axiom,
! [X3: joinTree_nat] :
( ! [Rel2: nat] :
( X3
!= ( relation_nat @ Rel2 ) )
=> ~ ! [L2: joinTree_nat,R: joinTree_nat] :
( X3
!= ( join_nat @ L2 @ R ) ) ) ).
% relations_mset.cases
thf(fact_159_relations__mset_Ocases,axiom,
! [X3: joinTree_a] :
( ! [Rel2: a] :
( X3
!= ( relation_a @ Rel2 ) )
=> ~ ! [L2: joinTree_a,R: joinTree_a] :
( X3
!= ( join_a @ L2 @ R ) ) ) ).
% relations_mset.cases
thf(fact_160_right__deep_Ocases,axiom,
! [X3: joinTree_complex] :
( ! [Uu: complex] :
( X3
!= ( relation_complex @ Uu ) )
=> ( ! [Uv: complex,R: joinTree_complex] :
( X3
!= ( join_complex @ ( relation_complex @ Uv ) @ R ) )
=> ~ ! [Vb: joinTree_complex,Vc: joinTree_complex,Va: joinTree_complex] :
( X3
!= ( join_complex @ ( join_complex @ Vb @ Vc ) @ Va ) ) ) ) ).
% right_deep.cases
thf(fact_161_right__deep_Ocases,axiom,
! [X3: joinTree_nat] :
( ! [Uu: nat] :
( X3
!= ( relation_nat @ Uu ) )
=> ( ! [Uv: nat,R: joinTree_nat] :
( X3
!= ( join_nat @ ( relation_nat @ Uv ) @ R ) )
=> ~ ! [Vb: joinTree_nat,Vc: joinTree_nat,Va: joinTree_nat] :
( X3
!= ( join_nat @ ( join_nat @ Vb @ Vc ) @ Va ) ) ) ) ).
% right_deep.cases
thf(fact_162_right__deep_Ocases,axiom,
! [X3: joinTree_a] :
( ! [Uu: a] :
( X3
!= ( relation_a @ Uu ) )
=> ( ! [Uv: a,R: joinTree_a] :
( X3
!= ( join_a @ ( relation_a @ Uv ) @ R ) )
=> ~ ! [Vb: joinTree_a,Vc: joinTree_a,Va: joinTree_a] :
( X3
!= ( join_a @ ( join_a @ Vb @ Vc ) @ Va ) ) ) ) ).
% right_deep.cases
thf(fact_163_joinTree_Oexhaust,axiom,
! [Y: joinTree_complex] :
( ! [X12: complex] :
( Y
!= ( relation_complex @ X12 ) )
=> ~ ! [X212: joinTree_complex,X222: joinTree_complex] :
( Y
!= ( join_complex @ X212 @ X222 ) ) ) ).
% joinTree.exhaust
thf(fact_164_joinTree_Oexhaust,axiom,
! [Y: joinTree_nat] :
( ! [X12: nat] :
( Y
!= ( relation_nat @ X12 ) )
=> ~ ! [X212: joinTree_nat,X222: joinTree_nat] :
( Y
!= ( join_nat @ X212 @ X222 ) ) ) ).
% joinTree.exhaust
thf(fact_165_joinTree_Oexhaust,axiom,
! [Y: joinTree_a] :
( ! [X12: a] :
( Y
!= ( relation_a @ X12 ) )
=> ~ ! [X212: joinTree_a,X222: joinTree_a] :
( Y
!= ( join_a @ X212 @ X222 ) ) ) ).
% joinTree.exhaust
thf(fact_166_left__deep_Ocases,axiom,
! [X3: joinTree_complex] :
( ! [Uu: complex] :
( X3
!= ( relation_complex @ Uu ) )
=> ( ! [L2: joinTree_complex,Uv: complex] :
( X3
!= ( join_complex @ L2 @ ( relation_complex @ Uv ) ) )
=> ~ ! [V: joinTree_complex,Vb: joinTree_complex,Vc: joinTree_complex] :
( X3
!= ( join_complex @ V @ ( join_complex @ Vb @ Vc ) ) ) ) ) ).
% left_deep.cases
thf(fact_167_left__deep_Ocases,axiom,
! [X3: joinTree_nat] :
( ! [Uu: nat] :
( X3
!= ( relation_nat @ Uu ) )
=> ( ! [L2: joinTree_nat,Uv: nat] :
( X3
!= ( join_nat @ L2 @ ( relation_nat @ Uv ) ) )
=> ~ ! [V: joinTree_nat,Vb: joinTree_nat,Vc: joinTree_nat] :
( X3
!= ( join_nat @ V @ ( join_nat @ Vb @ Vc ) ) ) ) ) ).
% left_deep.cases
thf(fact_168_left__deep_Ocases,axiom,
! [X3: joinTree_a] :
( ! [Uu: a] :
( X3
!= ( relation_a @ Uu ) )
=> ( ! [L2: joinTree_a,Uv: a] :
( X3
!= ( join_a @ L2 @ ( relation_a @ Uv ) ) )
=> ~ ! [V: joinTree_a,Vb: joinTree_a,Vc: joinTree_a] :
( X3
!= ( join_a @ V @ ( join_a @ Vb @ Vc ) ) ) ) ) ).
% left_deep.cases
thf(fact_169_ldeep__n__eq__card__subtree,axiom,
! [T: joinTree_complex,R3: joinTree_complex,F: complex > complex > real,Cf: complex > real] :
( ( distin1760132951325961430omplex @ ( join_complex @ T @ R3 ) )
=> ( ( left_deep_complex @ T )
=> ( ( ldeep_n_complex @ F @ Cf @ ( revorder_complex @ T ) )
= ( card_complex @ Cf @ F @ T ) ) ) ) ).
% ldeep_n_eq_card_subtree
thf(fact_170_ldeep__n__eq__card__subtree,axiom,
! [T: joinTree_nat,R3: joinTree_nat,F: nat > nat > real,Cf: nat > real] :
( ( distin9127876880396718072ns_nat @ ( join_nat @ T @ R3 ) )
=> ( ( left_deep_nat @ T )
=> ( ( ldeep_n_nat @ F @ Cf @ ( revorder_nat @ T ) )
= ( card_nat @ Cf @ F @ T ) ) ) ) ).
% ldeep_n_eq_card_subtree
thf(fact_171_ldeep__n__eq__card__subtree,axiom,
! [T: joinTree_a,R3: joinTree_a,F: a > a > real,Cf: a > real] :
( ( distinct_relations_a @ ( join_a @ T @ R3 ) )
=> ( ( left_deep_a @ T )
=> ( ( ldeep_n_a @ F @ Cf @ ( revorder_a @ T ) )
= ( card_a @ Cf @ F @ T ) ) ) ) ).
% ldeep_n_eq_card_subtree
thf(fact_172_ldeep__T__eq__card,axiom,
! [T: joinTree_a,F: a > a > real,Cf: a > real] :
( ( distinct_relations_a @ T )
=> ( ( left_deep_a @ T )
=> ( ( ldeep_T_a @ ( ldeep_s_a @ F @ ( revorder_a @ T ) ) @ Cf @ ( revorder_a @ T ) )
= ( card_a @ Cf @ F @ T ) ) ) ) ).
% ldeep_T_eq_card
thf(fact_173_ldeep__n__eq__card,axiom,
! [T: joinTree_a,F: a > a > real,Cf: a > real] :
( ( distinct_relations_a @ T )
=> ( ( left_deep_a @ T )
=> ( ( ldeep_n_a @ F @ Cf @ ( revorder_a @ T ) )
= ( card_a @ Cf @ F @ T ) ) ) ) ).
% ldeep_n_eq_card
thf(fact_174_c__nlj_Osimps_I2_J,axiom,
! [Cf: complex > real,F: complex > complex > real,L: joinTree_complex,R2: joinTree_complex] :
( ( c_nlj_complex @ Cf @ F @ ( join_complex @ L @ R2 ) )
= ( plus_plus_real @ ( plus_plus_real @ ( times_times_real @ ( card_complex @ Cf @ F @ L ) @ ( card_complex @ Cf @ F @ R2 ) ) @ ( c_nlj_complex @ Cf @ F @ L ) ) @ ( c_nlj_complex @ Cf @ F @ R2 ) ) ) ).
% c_nlj.simps(2)
thf(fact_175_c__nlj_Osimps_I2_J,axiom,
! [Cf: nat > real,F: nat > nat > real,L: joinTree_nat,R2: joinTree_nat] :
( ( c_nlj_nat @ Cf @ F @ ( join_nat @ L @ R2 ) )
= ( plus_plus_real @ ( plus_plus_real @ ( times_times_real @ ( card_nat @ Cf @ F @ L ) @ ( card_nat @ Cf @ F @ R2 ) ) @ ( c_nlj_nat @ Cf @ F @ L ) ) @ ( c_nlj_nat @ Cf @ F @ R2 ) ) ) ).
% c_nlj.simps(2)
thf(fact_176_c__nlj_Osimps_I2_J,axiom,
! [Cf: a > real,F: a > a > real,L: joinTree_a,R2: joinTree_a] :
( ( c_nlj_a @ Cf @ F @ ( join_a @ L @ R2 ) )
= ( plus_plus_real @ ( plus_plus_real @ ( times_times_real @ ( card_a @ Cf @ F @ L ) @ ( card_a @ Cf @ F @ R2 ) ) @ ( c_nlj_a @ Cf @ F @ L ) ) @ ( c_nlj_a @ Cf @ F @ R2 ) ) ) ).
% c_nlj.simps(2)
thf(fact_177_crossproduct__noteq,axiom,
! [A: complex,B: complex,C: complex,D: complex] :
( ( ( A != B )
& ( C != D ) )
= ( ( plus_plus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ D ) )
!= ( plus_plus_complex @ ( times_times_complex @ A @ D ) @ ( times_times_complex @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_178_crossproduct__noteq,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( A != B )
& ( C != D ) )
= ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) )
!= ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_179_crossproduct__noteq,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( A != B )
& ( C != D ) )
= ( ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) )
!= ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_180_crossproduct__eq,axiom,
! [W: complex,Y: complex,X3: complex,Z: complex] :
( ( ( plus_plus_complex @ ( times_times_complex @ W @ Y ) @ ( times_times_complex @ X3 @ Z ) )
= ( plus_plus_complex @ ( times_times_complex @ W @ Z ) @ ( times_times_complex @ X3 @ Y ) ) )
= ( ( W = X3 )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_181_crossproduct__eq,axiom,
! [W: nat,Y: nat,X3: nat,Z: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y ) @ ( times_times_nat @ X3 @ Z ) )
= ( plus_plus_nat @ ( times_times_nat @ W @ Z ) @ ( times_times_nat @ X3 @ Y ) ) )
= ( ( W = X3 )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_182_crossproduct__eq,axiom,
! [W: real,Y: real,X3: real,Z: real] :
( ( ( plus_plus_real @ ( times_times_real @ W @ Y ) @ ( times_times_real @ X3 @ Z ) )
= ( plus_plus_real @ ( times_times_real @ W @ Z ) @ ( times_times_real @ X3 @ Y ) ) )
= ( ( W = X3 )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_183_combine__common__factor,axiom,
! [A: complex,E: complex,B: complex,C: complex] :
( ( plus_plus_complex @ ( times_times_complex @ A @ E ) @ ( plus_plus_complex @ ( times_times_complex @ B @ E ) @ C ) )
= ( plus_plus_complex @ ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_184_combine__common__factor,axiom,
! [A: nat,E: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_185_combine__common__factor,axiom,
! [A: real,E: real,B: real,C: real] :
( ( plus_plus_real @ ( times_times_real @ A @ E ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ C ) )
= ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_186_distrib__right,axiom,
! [A: complex,B: complex,C: complex] :
( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ C )
= ( plus_plus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) ) ) ).
% distrib_right
thf(fact_187_distrib__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% distrib_right
thf(fact_188_distrib__right,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% distrib_right
thf(fact_189_distrib__left,axiom,
! [A: complex,B: complex,C: complex] :
( ( times_times_complex @ A @ ( plus_plus_complex @ B @ C ) )
= ( plus_plus_complex @ ( times_times_complex @ A @ B ) @ ( times_times_complex @ A @ C ) ) ) ).
% distrib_left
thf(fact_190_distrib__left,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% distrib_left
thf(fact_191_distrib__left,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
= ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% distrib_left
thf(fact_192_comm__semiring__class_Odistrib,axiom,
! [A: complex,B: complex,C: complex] :
( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ C )
= ( plus_plus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_193_comm__semiring__class_Odistrib,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_194_comm__semiring__class_Odistrib,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_195_ring__class_Oring__distribs_I1_J,axiom,
! [A: complex,B: complex,C: complex] :
( ( times_times_complex @ A @ ( plus_plus_complex @ B @ C ) )
= ( plus_plus_complex @ ( times_times_complex @ A @ B ) @ ( times_times_complex @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_196_ring__class_Oring__distribs_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
= ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_197_ring__class_Oring__distribs_I2_J,axiom,
! [A: complex,B: complex,C: complex] :
( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ C )
= ( plus_plus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_198_ring__class_Oring__distribs_I2_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_199_c__nlj_Oelims,axiom,
! [X3: complex > real,Xa: complex > complex > real,Xb: joinTree_complex,Y: real] :
( ( ( c_nlj_complex @ X3 @ Xa @ Xb )
= Y )
=> ( ( ? [Uw: complex] :
( Xb
= ( relation_complex @ Uw ) )
=> ( Y != zero_zero_real ) )
=> ~ ! [L2: joinTree_complex,R: joinTree_complex] :
( ( Xb
= ( join_complex @ L2 @ R ) )
=> ( Y
!= ( plus_plus_real @ ( plus_plus_real @ ( times_times_real @ ( card_complex @ X3 @ Xa @ L2 ) @ ( card_complex @ X3 @ Xa @ R ) ) @ ( c_nlj_complex @ X3 @ Xa @ L2 ) ) @ ( c_nlj_complex @ X3 @ Xa @ R ) ) ) ) ) ) ).
% c_nlj.elims
thf(fact_200_c__nlj_Oelims,axiom,
! [X3: nat > real,Xa: nat > nat > real,Xb: joinTree_nat,Y: real] :
( ( ( c_nlj_nat @ X3 @ Xa @ Xb )
= Y )
=> ( ( ? [Uw: nat] :
( Xb
= ( relation_nat @ Uw ) )
=> ( Y != zero_zero_real ) )
=> ~ ! [L2: joinTree_nat,R: joinTree_nat] :
( ( Xb
= ( join_nat @ L2 @ R ) )
=> ( Y
!= ( plus_plus_real @ ( plus_plus_real @ ( times_times_real @ ( card_nat @ X3 @ Xa @ L2 ) @ ( card_nat @ X3 @ Xa @ R ) ) @ ( c_nlj_nat @ X3 @ Xa @ L2 ) ) @ ( c_nlj_nat @ X3 @ Xa @ R ) ) ) ) ) ) ).
% c_nlj.elims
thf(fact_201_c__nlj_Oelims,axiom,
! [X3: a > real,Xa: a > a > real,Xb: joinTree_a,Y: real] :
( ( ( c_nlj_a @ X3 @ Xa @ Xb )
= Y )
=> ( ( ? [Uw: a] :
( Xb
= ( relation_a @ Uw ) )
=> ( Y != zero_zero_real ) )
=> ~ ! [L2: joinTree_a,R: joinTree_a] :
( ( Xb
= ( join_a @ L2 @ R ) )
=> ( Y
!= ( plus_plus_real @ ( plus_plus_real @ ( times_times_real @ ( card_a @ X3 @ Xa @ L2 ) @ ( card_a @ X3 @ Xa @ R ) ) @ ( c_nlj_a @ X3 @ Xa @ L2 ) ) @ ( c_nlj_a @ X3 @ Xa @ R ) ) ) ) ) ) ).
% c_nlj.elims
thf(fact_202_c__out_Osimps_I2_J,axiom,
! [Cf: a > real,F: a > a > real,L: joinTree_a,R2: joinTree_a] :
( ( c_out_a @ Cf @ F @ ( join_a @ L @ R2 ) )
= ( plus_plus_real @ ( plus_plus_real @ ( card_a @ Cf @ F @ ( join_a @ L @ R2 ) ) @ ( c_out_a @ Cf @ F @ L ) ) @ ( c_out_a @ Cf @ F @ R2 ) ) ) ).
% c_out.simps(2)
thf(fact_203_card__join__alt,axiom,
! [Cf: a > real,F: a > a > real,L: joinTree_a,R2: joinTree_a] :
( ( card_a @ Cf @ F @ ( join_a @ L @ R2 ) )
= ( times_times_real @ ( times_times_real @ ( list_sel_a @ F @ ( revorder_a @ L ) @ ( revorder_a @ R2 ) ) @ ( card_a @ Cf @ F @ L ) ) @ ( card_a @ Cf @ F @ R2 ) ) ) ).
% card_join_alt
thf(fact_204_distinct__c__IKKBZ__ldeep__s__subtree,axiom,
! [L: joinTree_a,Rel: a,F: a > a > real,Cf: a > real] :
( ( distinct_relations_a @ ( join_a @ L @ ( relation_a @ Rel ) ) )
=> ( ( left_deep_a @ ( join_a @ L @ ( relation_a @ Rel ) ) )
=> ( ( c_IKKBZ_a
@ ^ [A2: a] : ( times_times_real @ ( ldeep_s_a @ F @ ( revorder_a @ ( join_a @ L @ ( relation_a @ Rel ) ) ) @ A2 ) )
@ Cf
@ F
@ L )
= ( c_IKKBZ_a
@ ^ [A2: a] : ( times_times_real @ ( ldeep_s_a @ F @ ( revorder_a @ L ) @ A2 ) )
@ Cf
@ F
@ L ) ) ) ) ).
% distinct_c_IKKBZ_ldeep_s_subtree
thf(fact_205_ldeep__T__eq__ldeep__n,axiom,
! [Xs: list_a,F: a > a > real,Cf: a > real] :
( ( distinct_a @ Xs )
=> ( ( ldeep_T_a @ ( ldeep_s_a @ F @ Xs ) @ Cf @ Xs )
= ( ldeep_n_a @ F @ Cf @ Xs ) ) ) ).
% ldeep_T_eq_ldeep_n
thf(fact_206_c__IKKBZ_Oelims,axiom,
! [X3: a > real > real,Xa: a > real,Xb: a > a > real,Xc: joinTree_a,Y: real] :
( ( ( c_IKKBZ_a @ X3 @ Xa @ Xb @ Xc )
= Y )
=> ( ( ? [Ux: a] :
( Xc
= ( relation_a @ Ux ) )
=> ( Y != zero_zero_real ) )
=> ( ! [L2: joinTree_a,Rel2: a] :
( ( Xc
= ( join_a @ L2 @ ( relation_a @ Rel2 ) ) )
=> ( Y
!= ( plus_plus_real @ ( times_times_real @ ( card_a @ Xa @ Xb @ L2 ) @ ( X3 @ Rel2 @ ( Xa @ Rel2 ) ) ) @ ( c_IKKBZ_a @ X3 @ Xa @ Xb @ L2 ) ) ) )
=> ~ ( ? [L2: joinTree_a,V: joinTree_a,Vb: joinTree_a] :
( Xc
= ( join_a @ L2 @ ( join_a @ V @ Vb ) ) )
=> ( Y != undefined_real ) ) ) ) ) ).
% c_IKKBZ.elims
thf(fact_207_symmetric__def,axiom,
( symmetric_a
= ( ^ [F2: joinTree_a > real] :
! [X4: joinTree_a,Y2: joinTree_a] :
( ( F2 @ ( join_a @ X4 @ Y2 ) )
= ( F2 @ ( join_a @ Y2 @ X4 ) ) ) ) ) ).
% symmetric_def
thf(fact_208_is__num__normalize_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_209_right__deep_Oelims_I3_J,axiom,
! [X3: joinTree_a] :
( ~ ( right_deep_a @ X3 )
=> ( ! [Uv: a,R: joinTree_a] :
( ( X3
= ( join_a @ ( relation_a @ Uv ) @ R ) )
=> ( right_deep_a @ R ) )
=> ~ ! [Vb: joinTree_a,Vc: joinTree_a,Va: joinTree_a] :
( X3
!= ( join_a @ ( join_a @ Vb @ Vc ) @ Va ) ) ) ) ).
% right_deep.elims(3)
thf(fact_210_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_211_mult__cancel__right,axiom,
! [A: complex,C: complex,B: complex] :
( ( ( times_times_complex @ A @ C )
= ( times_times_complex @ B @ C ) )
= ( ( C = zero_zero_complex )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_212_mult__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_213_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_214_mult__cancel__left,axiom,
! [C: complex,A: complex,B: complex] :
( ( ( times_times_complex @ C @ A )
= ( times_times_complex @ C @ B ) )
= ( ( C = zero_zero_complex )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_215_mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_216_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_217_mult__eq__0__iff,axiom,
! [A: complex,B: complex] :
( ( ( times_times_complex @ A @ B )
= zero_zero_complex )
= ( ( A = zero_zero_complex )
| ( B = zero_zero_complex ) ) ) ).
% mult_eq_0_iff
thf(fact_218_mult__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_219_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_220_mult__zero__right,axiom,
! [A: complex] :
( ( times_times_complex @ A @ zero_zero_complex )
= zero_zero_complex ) ).
% mult_zero_right
thf(fact_221_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_222_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_223_mult__zero__left,axiom,
! [A: complex] :
( ( times_times_complex @ zero_zero_complex @ A )
= zero_zero_complex ) ).
% mult_zero_left
thf(fact_224_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_225_add__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add_0
thf(fact_226_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_227_add__0,axiom,
! [A: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A )
= A ) ).
% add_0
thf(fact_228_zero__eq__add__iff__both__eq__0,axiom,
! [X3: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X3 @ Y ) )
= ( ( X3 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_229_add__eq__0__iff__both__eq__0,axiom,
! [X3: nat,Y: nat] :
( ( ( plus_plus_nat @ X3 @ Y )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_230_add__cancel__right__right,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ A @ B ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_right
thf(fact_231_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_232_add__cancel__right__right,axiom,
! [A: complex,B: complex] :
( ( A
= ( plus_plus_complex @ A @ B ) )
= ( B = zero_zero_complex ) ) ).
% add_cancel_right_right
thf(fact_233_add__cancel__right__left,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ B @ A ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_left
thf(fact_234_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_235_add__cancel__right__left,axiom,
! [A: complex,B: complex] :
( ( A
= ( plus_plus_complex @ B @ A ) )
= ( B = zero_zero_complex ) ) ).
% add_cancel_right_left
thf(fact_236_add__cancel__left__right,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_right
thf(fact_237_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_238_add__cancel__left__right,axiom,
! [A: complex,B: complex] :
( ( ( plus_plus_complex @ A @ B )
= A )
= ( B = zero_zero_complex ) ) ).
% add_cancel_left_right
thf(fact_239_add__cancel__left__left,axiom,
! [B: real,A: real] :
( ( ( plus_plus_real @ B @ A )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_left
thf(fact_240_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_241_add__cancel__left__left,axiom,
! [B: complex,A: complex] :
( ( ( plus_plus_complex @ B @ A )
= A )
= ( B = zero_zero_complex ) ) ).
% add_cancel_left_left
thf(fact_242_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_243_add_Oright__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.right_neutral
thf(fact_244_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_245_add_Oright__neutral,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ zero_zero_complex )
= A ) ).
% add.right_neutral
thf(fact_246_lambda__zero,axiom,
( ( ^ [H2: nat] : zero_zero_nat )
= ( times_times_nat @ zero_zero_nat ) ) ).
% lambda_zero
thf(fact_247_lambda__zero,axiom,
( ( ^ [H2: complex] : zero_zero_complex )
= ( times_times_complex @ zero_zero_complex ) ) ).
% lambda_zero
thf(fact_248_lambda__zero,axiom,
( ( ^ [H2: real] : zero_zero_real )
= ( times_times_real @ zero_zero_real ) ) ).
% lambda_zero
thf(fact_249_zero__reorient,axiom,
! [X3: real] :
( ( zero_zero_real = X3 )
= ( X3 = zero_zero_real ) ) ).
% zero_reorient
thf(fact_250_zero__reorient,axiom,
! [X3: nat] :
( ( zero_zero_nat = X3 )
= ( X3 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_251_zero__reorient,axiom,
! [X3: complex] :
( ( zero_zero_complex = X3 )
= ( X3 = zero_zero_complex ) ) ).
% zero_reorient
thf(fact_252_c__out_Osimps_I1_J,axiom,
! [Uu2: a > real,Uv2: a > a > real,Uw2: a] :
( ( c_out_a @ Uu2 @ Uv2 @ ( relation_a @ Uw2 ) )
= zero_zero_real ) ).
% c_out.simps(1)
thf(fact_253_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_254_mult__right__cancel,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( ( times_times_complex @ A @ C )
= ( times_times_complex @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_255_mult__right__cancel,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_256_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_257_mult__left__cancel,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( ( times_times_complex @ C @ A )
= ( times_times_complex @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_258_mult__left__cancel,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_259_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_260_no__zero__divisors,axiom,
! [A: complex,B: complex] :
( ( A != zero_zero_complex )
=> ( ( B != zero_zero_complex )
=> ( ( times_times_complex @ A @ B )
!= zero_zero_complex ) ) ) ).
% no_zero_divisors
thf(fact_261_no__zero__divisors,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( times_times_real @ A @ B )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_262_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_263_divisors__zero,axiom,
! [A: complex,B: complex] :
( ( ( times_times_complex @ A @ B )
= zero_zero_complex )
=> ( ( A = zero_zero_complex )
| ( B = zero_zero_complex ) ) ) ).
% divisors_zero
thf(fact_264_divisors__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
=> ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_265_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_266_mult__not__zero,axiom,
! [A: complex,B: complex] :
( ( ( times_times_complex @ A @ B )
!= zero_zero_complex )
=> ( ( A != zero_zero_complex )
& ( B != zero_zero_complex ) ) ) ).
% mult_not_zero
thf(fact_267_mult__not__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
!= zero_zero_real )
=> ( ( A != zero_zero_real )
& ( B != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_268_add_Ogroup__left__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_269_add_Ogroup__left__neutral,axiom,
! [A: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A )
= A ) ).
% add.group_left_neutral
thf(fact_270_add_Ocomm__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.comm_neutral
thf(fact_271_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_272_add_Ocomm__neutral,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ zero_zero_complex )
= A ) ).
% add.comm_neutral
thf(fact_273_comm__monoid__add__class_Oadd__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_274_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_275_comm__monoid__add__class_Oadd__0,axiom,
! [A: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_276_add__0__iff,axiom,
! [B: real,A: real] :
( ( B
= ( plus_plus_real @ B @ A ) )
= ( A = zero_zero_real ) ) ).
% add_0_iff
thf(fact_277_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_278_add__0__iff,axiom,
! [B: complex,A: complex] :
( ( B
= ( plus_plus_complex @ B @ A ) )
= ( A = zero_zero_complex ) ) ).
% add_0_iff
thf(fact_279_right__deep_Osimps_I3_J,axiom,
! [Vb2: joinTree_a,Vc2: joinTree_a,Va2: joinTree_a] :
~ ( right_deep_a @ ( join_a @ ( join_a @ Vb2 @ Vc2 ) @ Va2 ) ) ).
% right_deep.simps(3)
thf(fact_280_right__deep_Osimps_I1_J,axiom,
! [Uu2: a] : ( right_deep_a @ ( relation_a @ Uu2 ) ) ).
% right_deep.simps(1)
thf(fact_281_c__IKKBZ__fun__notelem,axiom,
! [T: joinTree_a,Y: a,F3: a > real > real,Z: real > real,F: a > real > real,Cf: a > real,Sf: a > a > real] :
( ( left_deep_a @ T )
=> ( ( distinct_relations_a @ T )
=> ( ~ ( member_a @ Y @ ( relations_a @ T ) )
=> ( ( F3
= ( ^ [A2: a,B2: real] : ( if_real @ ( A2 = Y ) @ ( Z @ B2 ) @ ( F @ A2 @ B2 ) ) ) )
=> ( ( c_IKKBZ_a @ F3 @ Cf @ Sf @ T )
= ( c_IKKBZ_a @ F @ Cf @ Sf @ T ) ) ) ) ) ) ).
% c_IKKBZ_fun_notelem
thf(fact_282_add__scale__eq__noteq,axiom,
! [R2: nat,A: nat,B: nat,C: nat,D: nat] :
( ( R2 != zero_zero_nat )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_nat @ A @ ( times_times_nat @ R2 @ C ) )
!= ( plus_plus_nat @ B @ ( times_times_nat @ R2 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_283_add__scale__eq__noteq,axiom,
! [R2: complex,A: complex,B: complex,C: complex,D: complex] :
( ( R2 != zero_zero_complex )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_complex @ A @ ( times_times_complex @ R2 @ C ) )
!= ( plus_plus_complex @ B @ ( times_times_complex @ R2 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_284_add__scale__eq__noteq,axiom,
! [R2: real,A: real,B: real,C: real,D: real] :
( ( R2 != zero_zero_real )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_real @ A @ ( times_times_real @ R2 @ C ) )
!= ( plus_plus_real @ B @ ( times_times_real @ R2 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_285_distinct__rels__alt,axiom,
( distinct_relations_a
= ( ^ [T2: joinTree_a] : ( distinct_a @ ( revorder_a @ T2 ) ) ) ) ).
% distinct_rels_alt
thf(fact_286_c__out_Oelims,axiom,
! [X3: a > real,Xa: a > a > real,Xb: joinTree_a,Y: real] :
( ( ( c_out_a @ X3 @ Xa @ Xb )
= Y )
=> ( ( ? [Uw: a] :
( Xb
= ( relation_a @ Uw ) )
=> ( Y != zero_zero_real ) )
=> ~ ! [L2: joinTree_a,R: joinTree_a] :
( ( Xb
= ( join_a @ L2 @ R ) )
=> ( Y
!= ( plus_plus_real @ ( plus_plus_real @ ( card_a @ X3 @ Xa @ ( join_a @ L2 @ R ) ) @ ( c_out_a @ X3 @ Xa @ L2 ) ) @ ( c_out_a @ X3 @ Xa @ R ) ) ) ) ) ) ).
% c_out.elims
thf(fact_287_c__IKKBZ_Osimps_I1_J,axiom,
! [Uu2: a > real > real,Uv2: a > real,Uw2: a > a > real,Ux2: a] :
( ( c_IKKBZ_a @ Uu2 @ Uv2 @ Uw2 @ ( relation_a @ Ux2 ) )
= zero_zero_real ) ).
% c_IKKBZ.simps(1)
thf(fact_288_c__out__IKKBZ,axiom,
! [T: joinTree_a,Cf: a > real,F: a > a > real] :
( ( distinct_relations_a @ T )
=> ( ( reasonable_cards_a @ Cf @ F @ T )
=> ( ( left_deep_a @ T )
=> ( ( c_IKKBZ_a
@ ^ [A2: a] : ( times_times_real @ ( ldeep_s_a @ F @ ( revorder_a @ T ) @ A2 ) )
@ Cf
@ F
@ T )
= ( c_out_a @ Cf @ F @ T ) ) ) ) ) ).
% c_out_IKKBZ
thf(fact_289_c__IKKBZ_Osimps_I3_J,axiom,
! [Uy: a > real > real,Uz: a > real,Va2: a > a > real,L: joinTree_a,V2: joinTree_a,Vb2: joinTree_a] :
( ( c_IKKBZ_a @ Uy @ Uz @ Va2 @ ( join_a @ L @ ( join_a @ V2 @ Vb2 ) ) )
= undefined_real ) ).
% c_IKKBZ.simps(3)
thf(fact_290_c__nlj_Osimps_I1_J,axiom,
! [Uu2: a > real,Uv2: a > a > real,Uw2: a] :
( ( c_nlj_a @ Uu2 @ Uv2 @ ( relation_a @ Uw2 ) )
= zero_zero_real ) ).
% c_nlj.simps(1)
thf(fact_291_right__deep_Osimps_I2_J,axiom,
! [Uv2: a,R2: joinTree_a] :
( ( right_deep_a @ ( join_a @ ( relation_a @ Uv2 ) @ R2 ) )
= ( right_deep_a @ R2 ) ) ).
% right_deep.simps(2)
thf(fact_292_right__deep_Oelims_I1_J,axiom,
! [X3: joinTree_a,Y: $o] :
( ( ( right_deep_a @ X3 )
= Y )
=> ( ( ? [Uu: a] :
( X3
= ( relation_a @ Uu ) )
=> ~ Y )
=> ( ! [Uv: a,R: joinTree_a] :
( ( X3
= ( join_a @ ( relation_a @ Uv ) @ R ) )
=> ( Y
= ( ~ ( right_deep_a @ R ) ) ) )
=> ~ ( ? [Vb: joinTree_a,Vc: joinTree_a,Va: joinTree_a] :
( X3
= ( join_a @ ( join_a @ Vb @ Vc ) @ Va ) )
=> Y ) ) ) ) ).
% right_deep.elims(1)
thf(fact_293_right__deep_Oelims_I2_J,axiom,
! [X3: joinTree_a] :
( ( right_deep_a @ X3 )
=> ( ! [Uu: a] :
( X3
!= ( relation_a @ Uu ) )
=> ~ ! [Uv: a,R: joinTree_a] :
( ( X3
= ( join_a @ ( relation_a @ Uv ) @ R ) )
=> ~ ( right_deep_a @ R ) ) ) ) ).
% right_deep.elims(2)
thf(fact_294_sum__squares__eq__zero__iff,axiom,
! [X3: real,Y: real] :
( ( ( plus_plus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y @ Y ) )
= zero_zero_real )
= ( ( X3 = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_295_double__eq__0__iff,axiom,
! [A: real] :
( ( ( plus_plus_real @ A @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% double_eq_0_iff
thf(fact_296_distinct__c__IKKBZ__ldeep__s__prepend,axiom,
! [Ys: list_a,T: joinTree_a,F: a > a > real,Cf: a > real] :
( ( distinct_a @ ( append_a @ Ys @ ( revorder_a @ T ) ) )
=> ( ( left_deep_a @ T )
=> ( ( c_IKKBZ_a
@ ^ [A2: a] : ( times_times_real @ ( ldeep_s_a @ F @ ( append_a @ Ys @ ( revorder_a @ T ) ) @ A2 ) )
@ Cf
@ F
@ T )
= ( c_IKKBZ_a
@ ^ [A2: a] : ( times_times_real @ ( ldeep_s_a @ F @ ( revorder_a @ T ) @ A2 ) )
@ Cf
@ F
@ T ) ) ) ) ).
% distinct_c_IKKBZ_ldeep_s_prepend
thf(fact_297_verit__sum__simplify,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% verit_sum_simplify
thf(fact_298_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_299_verit__sum__simplify,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ zero_zero_complex )
= A ) ).
% verit_sum_simplify
thf(fact_300_card_Oelims,axiom,
! [X3: a > real,Xa: a > a > real,Xb: joinTree_a,Y: real] :
( ( ( card_a @ X3 @ Xa @ Xb )
= Y )
=> ( ! [Rel2: a] :
( ( Xb
= ( relation_a @ Rel2 ) )
=> ( Y
!= ( X3 @ Rel2 ) ) )
=> ~ ! [L2: joinTree_a,R: joinTree_a] :
( ( Xb
= ( join_a @ L2 @ R ) )
=> ( Y
!= ( times_times_real @ ( times_times_real @ ( list_sel_a @ Xa @ ( inorder_a @ L2 ) @ ( inorder_a @ R ) ) @ ( card_a @ X3 @ Xa @ L2 ) ) @ ( card_a @ X3 @ Xa @ R ) ) ) ) ) ) ).
% card.elims
thf(fact_301_ldeep__T_H__eq__ldeep__n,axiom,
! [Xs: list_a,F: a > a > real,Cf: a > real] :
( ( distinct_a @ Xs )
=> ( ( ldeep_T_a2 @ ( ldeep_s_a @ F @ Xs ) @ Cf @ Xs )
= ( ldeep_n_a @ F @ Cf @ Xs ) ) ) ).
% ldeep_T'_eq_ldeep_n
thf(fact_302_c__out__eq__c__list_H,axiom,
! [T: joinTree_a,Cf: a > real,F: a > a > real] :
( ( distinct_relations_a @ T )
=> ( ( reasonable_cards_a @ Cf @ F @ T )
=> ( ( left_deep_a @ T )
=> ( ( c_list_a2 @ F @ Cf
@ ^ [Xs2: list_a,X4: a] : ( times_times_real @ ( list_sel_aux_a @ F @ Xs2 @ X4 ) @ ( Cf @ X4 ) )
@ ( revorder_a @ T ) )
= ( c_out_a @ Cf @ F @ T ) ) ) ) ) ).
% c_out_eq_c_list'
thf(fact_303_card_Osimps_I2_J,axiom,
! [Cf: a > real,F: a > a > real,L: joinTree_a,R2: joinTree_a] :
( ( card_a @ Cf @ F @ ( join_a @ L @ R2 ) )
= ( times_times_real @ ( times_times_real @ ( list_sel_a @ F @ ( inorder_a @ L ) @ ( inorder_a @ R2 ) ) @ ( card_a @ Cf @ F @ L ) ) @ ( card_a @ Cf @ F @ R2 ) ) ) ).
% card.simps(2)
thf(fact_304_mult__commute__abs,axiom,
! [C: real] :
( ( ^ [X4: real] : ( times_times_real @ X4 @ C ) )
= ( times_times_real @ C ) ) ).
% mult_commute_abs
thf(fact_305_inorder_Osimps_I2_J,axiom,
! [L: joinTree_a,R2: joinTree_a] :
( ( inorder_a @ ( join_a @ L @ R2 ) )
= ( append_a @ ( inorder_a @ L ) @ ( inorder_a @ R2 ) ) ) ).
% inorder.simps(2)
thf(fact_306_distinct__ldeep__T_H__prepend,axiom,
! [Ys: list_a,Xs: list_a,F: a > a > real,Cf: a > real] :
( ( distinct_a @ ( append_a @ Ys @ Xs ) )
=> ( ( ldeep_T_a2 @ ( ldeep_s_a @ F @ ( append_a @ Ys @ Xs ) ) @ Cf @ Xs )
= ( ldeep_T_a2 @ ( ldeep_s_a @ F @ Xs ) @ Cf @ Xs ) ) ) ).
% distinct_ldeep_T'_prepend
thf(fact_307_revorder_Osimps_I2_J,axiom,
! [L: joinTree_a,R2: joinTree_a] :
( ( revorder_a @ ( join_a @ L @ R2 ) )
= ( append_a @ ( revorder_a @ R2 ) @ ( revorder_a @ L ) ) ) ).
% revorder.simps(2)
thf(fact_308_distinct__relations__def,axiom,
( distinct_relations_a
= ( ^ [T2: joinTree_a] : ( distinct_a @ ( inorder_a @ T2 ) ) ) ) ).
% distinct_relations_def
thf(fact_309_list__sel__revorder__eq__inorder__y,axiom,
! [F: a > a > real,Xs: list_a,R2: joinTree_a] :
( ( list_sel_a @ F @ Xs @ ( revorder_a @ R2 ) )
= ( list_sel_a @ F @ Xs @ ( inorder_a @ R2 ) ) ) ).
% list_sel_revorder_eq_inorder_y
thf(fact_310_list__sel__revorder__eq__inorder__x,axiom,
! [F: a > a > real,L: joinTree_a,Ys: list_a] :
( ( list_sel_a @ F @ ( revorder_a @ L ) @ Ys )
= ( list_sel_a @ F @ ( inorder_a @ L ) @ Ys ) ) ).
% list_sel_revorder_eq_inorder_x
thf(fact_311_list__sel__revorder__eq__inorder,axiom,
! [F: a > a > real,L: joinTree_a,R2: joinTree_a] :
( ( list_sel_a @ F @ ( revorder_a @ L ) @ ( revorder_a @ R2 ) )
= ( list_sel_a @ F @ ( inorder_a @ L ) @ ( inorder_a @ R2 ) ) ) ).
% list_sel_revorder_eq_inorder
thf(fact_312_c__list__prepend__f__disjunct,axiom,
! [Ys: list_a,Xs: list_a,F: a > a > real,Cf: a > real,H: a > real,R2: a] :
( ( distinct_a @ ( append_a @ Ys @ Xs ) )
=> ( ( c_list_a @ ( ldeep_s_a @ F @ ( append_a @ Ys @ Xs ) ) @ Cf @ H @ R2 @ Xs )
= ( c_list_a @ ( ldeep_s_a @ F @ Xs ) @ Cf @ H @ R2 @ Xs ) ) ) ).
% c_list_prepend_f_disjunct
thf(fact_313_distinct__ldeep__T__prepend,axiom,
! [Ys: list_a,Xs: list_a,F: a > a > real,Cf: a > real] :
( ( distinct_a @ ( append_a @ Ys @ Xs ) )
=> ( ( ldeep_T_a @ ( ldeep_s_a @ F @ ( append_a @ Ys @ Xs ) ) @ Cf @ Xs )
= ( ldeep_T_a @ ( ldeep_s_a @ F @ Xs ) @ Cf @ Xs ) ) ) ).
% distinct_ldeep_T_prepend
thf(fact_314_symmetric_H__def,axiom,
( symmetric_a2
= ( ^ [F2: ( a > real ) > ( a > a > real ) > joinTree_a > real] :
! [X4: joinTree_a,Y2: joinTree_a,Cf2: a > real,Sf2: a > a > real] :
( ( sel_symm_a @ Sf2 )
=> ( ( F2 @ Cf2 @ Sf2 @ ( join_a @ X4 @ Y2 ) )
= ( F2 @ Cf2 @ Sf2 @ ( join_a @ Y2 @ X4 ) ) ) ) ) ) ).
% symmetric'_def
thf(fact_315_ldeep__s__h__eq__create__h__list_H_H_H,axiom,
! [T: joinTree_a,X3: a,F: a > a > real,Cf: a > real] :
( ( distinct_relations_a @ T )
=> ( ( member_a @ X3 @ ( relations_a @ T ) )
=> ( ( times_times_real @ ( ldeep_s_a @ F @ ( revorder_a @ T ) @ X3 ) @ ( Cf @ X3 ) )
= ( create_h_list_a
@ ^ [Xs2: list_a,X4: a] : ( times_times_real @ ( list_sel_aux_a @ F @ Xs2 @ X4 ) @ ( Cf @ X4 ) )
@ ( revorder_a @ T )
@ X3 ) ) ) ) ).
% ldeep_s_h_eq_create_h_list'''
thf(fact_316_create__ldeep__inorder,axiom,
! [T: joinTree_a] :
( ( left_deep_a @ T )
=> ( ( create_ldeep_a @ ( inorder_a @ T ) )
= T ) ) ).
% create_ldeep_inorder
thf(fact_317_c__list__prepend__h__disjunct,axiom,
! [Ys: list_a,Xs: list_a,F: a > real,Cf: a > real,H: list_a > a > real,R2: a] :
( ( distinct_a @ ( append_a @ Ys @ Xs ) )
=> ( ( c_list_a @ F @ Cf @ ( create_h_list_a @ H @ ( append_a @ Ys @ Xs ) ) @ R2 @ Xs )
= ( c_list_a @ F @ Cf @ ( create_h_list_a @ H @ Xs ) @ R2 @ Xs ) ) ) ).
% c_list_prepend_h_disjunct
thf(fact_318_ldeep__s__h__eq__create__h__list_H_H,axiom,
! [T: joinTree_a,F: a > a > real,Cf: a > real] :
( ( distinct_relations_a @ T )
=> ! [Ys2: list_a,X: a,Zs: list_a] :
( ( ( append_a @ Ys2 @ ( cons_a @ X @ Zs ) )
= ( revorder_a @ T ) )
=> ( ( times_times_real @ ( ldeep_s_a @ F @ ( revorder_a @ T ) @ X ) @ ( Cf @ X ) )
= ( create_h_list_a
@ ^ [Xs2: list_a,Y2: a] : ( times_times_real @ ( list_sel_aux_a @ F @ Xs2 @ Y2 ) @ ( Cf @ Y2 ) )
@ ( revorder_a @ T )
@ X ) ) ) ) ).
% ldeep_s_h_eq_create_h_list''
thf(fact_319_ldeep__s__h__eq__create__h__list_H,axiom,
! [T: joinTree_a,Ys: list_a,X3: a,Zs2: list_a,F: a > a > real,Cf: a > real] :
( ( distinct_relations_a @ T )
=> ( ( ( append_a @ Ys @ ( cons_a @ X3 @ Zs2 ) )
= ( revorder_a @ T ) )
=> ( ( times_times_real @ ( ldeep_s_a @ F @ ( revorder_a @ T ) @ X3 ) @ ( Cf @ X3 ) )
= ( create_h_list_a
@ ^ [Xs2: list_a,X4: a] : ( times_times_real @ ( list_sel_aux_a @ F @ Xs2 @ X4 ) @ ( Cf @ X4 ) )
@ ( revorder_a @ T )
@ X3 ) ) ) ) ).
% ldeep_s_h_eq_create_h_list'
thf(fact_320_ldeep__s__h__eq__create__h__list,axiom,
! [Xs: list_a,Ys: list_a,X3: a,Zs2: list_a,F: a > a > real,Cf: a > real] :
( ( distinct_a @ Xs )
=> ( ( ( append_a @ Ys @ ( cons_a @ X3 @ Zs2 ) )
= Xs )
=> ( ( times_times_real @ ( ldeep_s_a @ F @ Xs @ X3 ) @ ( Cf @ X3 ) )
= ( create_h_list_a
@ ^ [Xs2: list_a,X4: a] : ( times_times_real @ ( list_sel_aux_a @ F @ Xs2 @ X4 ) @ ( Cf @ X4 ) )
@ Xs
@ X3 ) ) ) ) ).
% ldeep_s_h_eq_create_h_list
thf(fact_321_ldeep__s__h__eq__list__sel__aux_H__h_H,axiom,
! [T: joinTree_a,Ys: list_a,X3: a,Zs2: list_a,F: a > a > real,Cf: a > real] :
( ( distinct_relations_a @ T )
=> ( ( ( append_a @ Ys @ ( cons_a @ X3 @ Zs2 ) )
= ( revorder_a @ T ) )
=> ( ( times_times_real @ ( ldeep_s_a @ F @ ( revorder_a @ T ) @ X3 ) @ ( Cf @ X3 ) )
= ( times_times_real @ ( list_sel_aux_a @ F @ Zs2 @ X3 ) @ ( Cf @ X3 ) ) ) ) ) ).
% ldeep_s_h_eq_list_sel_aux'_h'
thf(fact_322_ldeep__s__h__eq__list__sel__aux_H__h,axiom,
! [Xs: list_a,Ys: list_a,X3: a,Zs2: list_a,F: a > a > real,Cf: a > real] :
( ( distinct_a @ Xs )
=> ( ( ( append_a @ Ys @ ( cons_a @ X3 @ Zs2 ) )
= Xs )
=> ( ( times_times_real @ ( ldeep_s_a @ F @ Xs @ X3 ) @ ( Cf @ X3 ) )
= ( times_times_real @ ( list_sel_aux_a @ F @ Zs2 @ X3 ) @ ( Cf @ X3 ) ) ) ) ) ).
% ldeep_s_h_eq_list_sel_aux'_h
thf(fact_323_create__rdeep_Osimps_I3_J,axiom,
! [X3: a,V2: a,Va2: list_a] :
( ( create_rdeep_a @ ( cons_a @ X3 @ ( cons_a @ V2 @ Va2 ) ) )
= ( join_a @ ( relation_a @ X3 ) @ ( create_rdeep_a @ ( cons_a @ V2 @ Va2 ) ) ) ) ).
% create_rdeep.simps(3)
thf(fact_324_ldeep__s_Osimps_I2_J,axiom,
! [F: a > a > real,X3: a,Xs: list_a] :
( ( ldeep_s_a @ F @ ( cons_a @ X3 @ Xs ) )
= ( ^ [A2: a] : ( if_real @ ( A2 = X3 ) @ ( list_sel_aux_a @ F @ Xs @ A2 ) @ ( ldeep_s_a @ F @ Xs @ A2 ) ) ) ) ).
% ldeep_s.simps(2)
thf(fact_325_first__node__first__inorder,axiom,
! [T: joinTree_a] :
? [Xs3: list_a] :
( ( inorder_a @ T )
= ( cons_a @ ( first_node_a @ T ) @ Xs3 ) ) ).
% first_node_first_inorder
thf(fact_326_distinct__ldeep__s__eq__aux_H,axiom,
! [Xs: list_a,As: list_a,Y: a,Bs: list_a,Sel: a > a > real] :
( ( distinct_a @ Xs )
=> ( ( ( append_a @ As @ ( cons_a @ Y @ Bs ) )
= Xs )
=> ( ( ldeep_s_a @ Sel @ Xs @ Y )
= ( list_sel_aux_a @ Sel @ Bs @ Y ) ) ) ) ).
% distinct_ldeep_s_eq_aux'
thf(fact_327_distinct__ldeep__s__eq__aux,axiom,
! [Xs: list_a,Y: a,Ys: list_a,F: a > a > real] :
( ( distinct_a @ Xs )
=> ( ? [Xs4: list_a] :
( ( append_a @ Xs4 @ ( cons_a @ Y @ Ys ) )
= Xs )
=> ( ( ldeep_s_a @ F @ Xs @ Y )
= ( list_sel_aux_a @ F @ Ys @ Y ) ) ) ) ).
% distinct_ldeep_s_eq_aux
thf(fact_328_c__list_Oelims,axiom,
! [X3: a > real,Xa: a > real,Xb: a > real,Xc: a,Xd: list_a,Y: real] :
( ( ( c_list_a @ X3 @ Xa @ Xb @ Xc @ Xd )
= Y )
=> ( ( ( Xd = nil_a )
=> ( Y != zero_zero_real ) )
=> ( ! [X2: a] :
( ( Xd
= ( cons_a @ X2 @ nil_a ) )
=> ~ ( ( ( X2 = Xc )
=> ( Y = zero_zero_real ) )
& ( ( X2 != Xc )
=> ( Y
= ( Xb @ X2 ) ) ) ) )
=> ~ ! [X2: a,V: a,Va: list_a] :
( ( Xd
= ( cons_a @ X2 @ ( cons_a @ V @ Va ) ) )
=> ( Y
!= ( plus_plus_real @ ( c_list_a @ X3 @ Xa @ Xb @ Xc @ ( cons_a @ V @ Va ) ) @ ( times_times_real @ ( ldeep_T_a @ X3 @ Xa @ ( cons_a @ V @ Va ) ) @ ( c_list_a @ X3 @ Xa @ Xb @ Xc @ ( cons_a @ X2 @ nil_a ) ) ) ) ) ) ) ) ) ).
% c_list.elims
thf(fact_329_c__list_H__eq__c__list,axiom,
! [Xs: list_a,R2: a,Rs: list_a,F: a > a > real,Cf: a > real,H: list_a > a > real] :
( ( distinct_a @ Xs )
=> ( ( ( rev_a @ Xs )
= ( cons_a @ R2 @ Rs ) )
=> ( ( c_list_a @ ( ldeep_s_a @ F @ Xs ) @ Cf @ ( create_h_list_a @ H @ Xs ) @ R2 @ Xs )
= ( c_list_a2 @ F @ Cf @ H @ Xs ) ) ) ) ).
% c_list'_eq_c_list
thf(fact_330_create__ldeep__snoc,axiom,
! [Xs: list_a,X3: a] :
( ( Xs != nil_a )
=> ( ( create_ldeep_a @ ( append_a @ Xs @ ( cons_a @ X3 @ nil_a ) ) )
= ( join_a @ ( create_ldeep_a @ Xs ) @ ( relation_a @ X3 ) ) ) ) ).
% create_ldeep_snoc
thf(fact_331_c__list_Osimps_I3_J,axiom,
! [Sf: a > real,Cf: a > real,H: a > real,R2: a,X3: a,V2: a,Va2: list_a] :
( ( c_list_a @ Sf @ Cf @ H @ R2 @ ( cons_a @ X3 @ ( cons_a @ V2 @ Va2 ) ) )
= ( plus_plus_real @ ( c_list_a @ Sf @ Cf @ H @ R2 @ ( cons_a @ V2 @ Va2 ) ) @ ( times_times_real @ ( ldeep_T_a @ Sf @ Cf @ ( cons_a @ V2 @ Va2 ) ) @ ( c_list_a @ Sf @ Cf @ H @ R2 @ ( cons_a @ X3 @ nil_a ) ) ) ) ) ).
% c_list.simps(3)
thf(fact_332_inorder_Oelims,axiom,
! [X3: joinTree_a,Y: list_a] :
( ( ( inorder_a @ X3 )
= Y )
=> ( ! [Rel2: a] :
( ( X3
= ( relation_a @ Rel2 ) )
=> ( Y
!= ( cons_a @ Rel2 @ nil_a ) ) )
=> ~ ! [L2: joinTree_a,R: joinTree_a] :
( ( X3
= ( join_a @ L2 @ R ) )
=> ( Y
!= ( append_a @ ( inorder_a @ L2 ) @ ( inorder_a @ R ) ) ) ) ) ) ).
% inorder.elims
thf(fact_333_revorder__nempty,axiom,
! [T: joinTree_a] :
( ( revorder_a @ T )
!= nil_a ) ).
% revorder_nempty
thf(fact_334_inorder__eq__rev__revorder,axiom,
( inorder_a
= ( ^ [T2: joinTree_a] : ( rev_a @ ( revorder_a @ T2 ) ) ) ) ).
% inorder_eq_rev_revorder
thf(fact_335_revorder__eq__rev__inorder,axiom,
( revorder_a
= ( ^ [T2: joinTree_a] : ( rev_a @ ( inorder_a @ T2 ) ) ) ) ).
% revorder_eq_rev_inorder
thf(fact_336_c__list_Osimps_I1_J,axiom,
! [Uu2: a > real,Uv2: a > real,Uw2: a > real,Ux2: a] :
( ( c_list_a @ Uu2 @ Uv2 @ Uw2 @ Ux2 @ nil_a )
= zero_zero_real ) ).
% c_list.simps(1)
thf(fact_337_create__ldeep__ldeep,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( left_deep_a @ ( create_ldeep_a @ Xs ) ) ) ).
% create_ldeep_ldeep
thf(fact_338_revorder_Osimps_I1_J,axiom,
! [Rel: a] :
( ( revorder_a @ ( relation_a @ Rel ) )
= ( cons_a @ Rel @ nil_a ) ) ).
% revorder.simps(1)
thf(fact_339_inorder_Osimps_I1_J,axiom,
! [Rel: a] :
( ( inorder_a @ ( relation_a @ Rel ) )
= ( cons_a @ Rel @ nil_a ) ) ).
% inorder.simps(1)
thf(fact_340_c__list_Osimps_I2_J,axiom,
! [X3: a,R2: a,Uy: a > real,Uz: a > real,H: a > real] :
( ( ( X3 = R2 )
=> ( ( c_list_a @ Uy @ Uz @ H @ R2 @ ( cons_a @ X3 @ nil_a ) )
= zero_zero_real ) )
& ( ( X3 != R2 )
=> ( ( c_list_a @ Uy @ Uz @ H @ R2 @ ( cons_a @ X3 @ nil_a ) )
= ( H @ X3 ) ) ) ) ).
% c_list.simps(2)
thf(fact_341_create__rdeep_Osimps_I2_J,axiom,
! [X3: a] :
( ( create_rdeep_a @ ( cons_a @ X3 @ nil_a ) )
= ( relation_a @ X3 ) ) ).
% create_rdeep.simps(2)
thf(fact_342_first__node__last__revorder,axiom,
! [T: joinTree_a] :
? [Xs3: list_a] :
( ( revorder_a @ T )
= ( append_a @ Xs3 @ ( cons_a @ ( first_node_a @ T ) @ nil_a ) ) ) ).
% first_node_last_revorder
thf(fact_343_revorder_Oelims,axiom,
! [X3: joinTree_a,Y: list_a] :
( ( ( revorder_a @ X3 )
= Y )
=> ( ! [Rel2: a] :
( ( X3
= ( relation_a @ Rel2 ) )
=> ( Y
!= ( cons_a @ Rel2 @ nil_a ) ) )
=> ~ ! [L2: joinTree_a,R: joinTree_a] :
( ( X3
= ( join_a @ L2 @ R ) )
=> ( Y
!= ( append_a @ ( revorder_a @ R ) @ ( revorder_a @ L2 ) ) ) ) ) ) ).
% revorder.elims
thf(fact_344_ldeep__div__eq__sel,axiom,
! [Cf: a > real,F: a > a > real,L: joinTree_a,Rel: a,C: real,Cr: real] :
( ( reasonable_cards_a @ Cf @ F @ ( join_a @ L @ ( relation_a @ Rel ) ) )
=> ( ( C
= ( card_a @ Cf @ F @ ( join_a @ L @ ( relation_a @ Rel ) ) ) )
=> ( ( Cr
= ( card_a @ Cf @ F @ ( relation_a @ Rel ) ) )
=> ( ( divide_divide_real @ C @ ( times_times_real @ ( card_a @ Cf @ F @ L ) @ Cr ) )
= ( list_sel_a @ F @ ( inorder_a @ L ) @ ( cons_a @ Rel @ nil_a ) ) ) ) ) ) ).
% ldeep_div_eq_sel
thf(fact_345_create__ldeep__rev__Cons,axiom,
! [Xs: list_a,X3: a] :
( ( Xs != nil_a )
=> ( ( create_ldeep_rev_a @ ( cons_a @ X3 @ Xs ) )
= ( join_a @ ( create_ldeep_rev_a @ Xs ) @ ( relation_a @ X3 ) ) ) ) ).
% create_ldeep_rev_Cons
thf(fact_346_create__rdeep_Oelims,axiom,
! [X3: list_a,Y: joinTree_a] :
( ( ( create_rdeep_a @ X3 )
= Y )
=> ( ( ( X3 = nil_a )
=> ( Y != undefined_joinTree_a ) )
=> ( ! [X2: a] :
( ( X3
= ( cons_a @ X2 @ nil_a ) )
=> ( Y
!= ( relation_a @ X2 ) ) )
=> ~ ! [X2: a,V: a,Va: list_a] :
( ( X3
= ( cons_a @ X2 @ ( cons_a @ V @ Va ) ) )
=> ( Y
!= ( join_a @ ( relation_a @ X2 ) @ ( create_rdeep_a @ ( cons_a @ V @ Va ) ) ) ) ) ) ) ) ).
% create_rdeep.elims
thf(fact_347_c__list__single__h__list__not__elem__prepend,axiom,
! [X3: a,Ys: list_a,F: a > real,Cf: a > real,H: list_a > a > real,Xs: list_a,R2: a] :
( ~ ( member_a @ X3 @ ( set_a2 @ Ys ) )
=> ( ( c_list_a @ F @ Cf @ ( create_h_list_a @ H @ ( append_a @ Ys @ ( cons_a @ X3 @ Xs ) ) ) @ R2 @ ( cons_a @ X3 @ nil_a ) )
= ( c_list_a @ F @ Cf @ ( create_h_list_a @ H @ ( cons_a @ X3 @ Xs ) ) @ R2 @ ( cons_a @ X3 @ nil_a ) ) ) ) ).
% c_list_single_h_list_not_elem_prepend
thf(fact_348_c__list__single__f__list__not__elem__prepend,axiom,
! [X3: a,Ys: list_a,F: a > a > real,Xs: list_a,Cf: a > real,H: a > real,R2: a] :
( ~ ( member_a @ X3 @ ( set_a2 @ Ys ) )
=> ( ( c_list_a @ ( ldeep_s_a @ F @ ( append_a @ Ys @ ( cons_a @ X3 @ Xs ) ) ) @ Cf @ H @ R2 @ ( cons_a @ X3 @ nil_a ) )
= ( c_list_a @ ( ldeep_s_a @ F @ ( cons_a @ X3 @ Xs ) ) @ Cf @ H @ R2 @ ( cons_a @ X3 @ nil_a ) ) ) ) ).
% c_list_single_f_list_not_elem_prepend
thf(fact_349_div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% div_0
thf(fact_350_div__0,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ zero_zero_complex @ A )
= zero_zero_complex ) ).
% div_0
thf(fact_351_div__0,axiom,
! [A: real] :
( ( divide_divide_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% div_0
thf(fact_352_div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_353_div__by__0,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
= zero_zero_complex ) ).
% div_by_0
thf(fact_354_div__by__0,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% div_by_0
thf(fact_355_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_356_mult_Oright__neutral,axiom,
! [A: complex] :
( ( times_times_complex @ A @ one_one_complex )
= A ) ).
% mult.right_neutral
thf(fact_357_mult_Oright__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.right_neutral
thf(fact_358_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_359_mult__1,axiom,
! [A: complex] :
( ( times_times_complex @ one_one_complex @ A )
= A ) ).
% mult_1
thf(fact_360_mult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% mult_1
thf(fact_361_div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% div_by_1
thf(fact_362_div__by__1,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ A @ one_one_complex )
= A ) ).
% div_by_1
thf(fact_363_div__by__1,axiom,
! [A: real] :
( ( divide_divide_real @ A @ one_one_real )
= A ) ).
% div_by_1
thf(fact_364_mult__cancel__right2,axiom,
! [A: complex,C: complex] :
( ( ( times_times_complex @ A @ C )
= C )
= ( ( C = zero_zero_complex )
| ( A = one_one_complex ) ) ) ).
% mult_cancel_right2
thf(fact_365_mult__cancel__right2,axiom,
! [A: real,C: real] :
( ( ( times_times_real @ A @ C )
= C )
= ( ( C = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_366_mult__cancel__right1,axiom,
! [C: complex,B: complex] :
( ( C
= ( times_times_complex @ B @ C ) )
= ( ( C = zero_zero_complex )
| ( B = one_one_complex ) ) ) ).
% mult_cancel_right1
thf(fact_367_mult__cancel__right1,axiom,
! [C: real,B: real] :
( ( C
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_368_mult__cancel__left2,axiom,
! [C: complex,A: complex] :
( ( ( times_times_complex @ C @ A )
= C )
= ( ( C = zero_zero_complex )
| ( A = one_one_complex ) ) ) ).
% mult_cancel_left2
thf(fact_369_mult__cancel__left2,axiom,
! [C: real,A: real] :
( ( ( times_times_real @ C @ A )
= C )
= ( ( C = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_370_mult__cancel__left1,axiom,
! [C: complex,B: complex] :
( ( C
= ( times_times_complex @ C @ B ) )
= ( ( C = zero_zero_complex )
| ( B = one_one_complex ) ) ) ).
% mult_cancel_left1
thf(fact_371_mult__cancel__left1,axiom,
! [C: real,B: real] :
( ( C
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_372_nonzero__mult__div__cancel__right,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_373_nonzero__mult__div__cancel__right,axiom,
! [B: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_374_nonzero__mult__div__cancel__right,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_375_nonzero__mult__div__cancel__left,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_376_nonzero__mult__div__cancel__left,axiom,
! [A: complex,B: complex] :
( ( A != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_377_nonzero__mult__div__cancel__left,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_378_div__self,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ A @ A )
= one_one_nat ) ) ).
% div_self
thf(fact_379_div__self,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ A )
= one_one_complex ) ) ).
% div_self
thf(fact_380_div__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% div_self
thf(fact_381_ldeep__s__1__if__nelem,axiom,
! [X3: a,Xs: list_a,Sel: a > a > real] :
( ~ ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( ( ldeep_s_a @ Sel @ Xs @ X3 )
= one_one_real ) ) ).
% ldeep_s_1_if_nelem
thf(fact_382_one__reorient,axiom,
! [X3: real] :
( ( one_one_real = X3 )
= ( X3 = one_one_real ) ) ).
% one_reorient
thf(fact_383_one__reorient,axiom,
! [X3: nat] :
( ( one_one_nat = X3 )
= ( X3 = one_one_nat ) ) ).
% one_reorient
thf(fact_384_one__reorient,axiom,
! [X3: complex] :
( ( one_one_complex = X3 )
= ( X3 = one_one_complex ) ) ).
% one_reorient
thf(fact_385_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_386_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_387_zero__neq__one,axiom,
zero_zero_complex != one_one_complex ).
% zero_neq_one
thf(fact_388_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_389_comm__monoid__mult__class_Omult__1,axiom,
! [A: complex] :
( ( times_times_complex @ one_one_complex @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_390_comm__monoid__mult__class_Omult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_391_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_392_mult_Ocomm__neutral,axiom,
! [A: complex] :
( ( times_times_complex @ A @ one_one_complex )
= A ) ).
% mult.comm_neutral
thf(fact_393_mult_Ocomm__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.comm_neutral
thf(fact_394_create__ldeep__rev__relations,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( relations_a @ ( create_ldeep_rev_a @ Xs ) )
= ( set_a2 @ Xs ) ) ) ).
% create_ldeep_rev_relations
thf(fact_395_lambda__one,axiom,
( ( ^ [X4: nat] : X4 )
= ( times_times_nat @ one_one_nat ) ) ).
% lambda_one
thf(fact_396_lambda__one,axiom,
( ( ^ [X4: complex] : X4 )
= ( times_times_complex @ one_one_complex ) ) ).
% lambda_one
thf(fact_397_lambda__one,axiom,
( ( ^ [X4: real] : X4 )
= ( times_times_real @ one_one_real ) ) ).
% lambda_one
thf(fact_398_revorder__eq__set,axiom,
! [T: joinTree_a] :
( ( set_a2 @ ( revorder_a @ T ) )
= ( relations_a @ T ) ) ).
% revorder_eq_set
thf(fact_399_inorder__eq__set,axiom,
! [T: joinTree_a] :
( ( set_a2 @ ( inorder_a @ T ) )
= ( relations_a @ T ) ) ).
% inorder_eq_set
thf(fact_400_ldeep__s_Osimps_I1_J,axiom,
! [F: a > a > real] :
( ( ldeep_s_a @ F @ nil_a )
= ( ^ [Uu3: a] : one_one_real ) ) ).
% ldeep_s.simps(1)
thf(fact_401_create__ldeep__rev_Oelims,axiom,
! [X3: list_a,Y: joinTree_a] :
( ( ( create_ldeep_rev_a @ X3 )
= Y )
=> ( ( ( X3 = nil_a )
=> ( Y != undefined_joinTree_a ) )
=> ( ! [X2: a] :
( ( X3
= ( cons_a @ X2 @ nil_a ) )
=> ( Y
!= ( relation_a @ X2 ) ) )
=> ~ ! [X2: a,V: a,Va: list_a] :
( ( X3
= ( cons_a @ X2 @ ( cons_a @ V @ Va ) ) )
=> ( Y
!= ( join_a @ ( create_ldeep_rev_a @ ( cons_a @ V @ Va ) ) @ ( relation_a @ X2 ) ) ) ) ) ) ) ).
% create_ldeep_rev.elims
thf(fact_402_create__ldeep__relations,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( relations_a @ ( create_ldeep_a @ Xs ) )
= ( set_a2 @ Xs ) ) ) ).
% create_ldeep_relations
thf(fact_403_create__ldeep__rev__ldeep,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( left_deep_a @ ( create_ldeep_rev_a @ Xs ) ) ) ).
% create_ldeep_rev_ldeep
thf(fact_404_ldeep__s__eq__list__sel__aux_H__split,axiom,
! [Y: a,Xs: list_a,Sel: a > a > real] :
( ( member_a @ Y @ ( set_a2 @ Xs ) )
=> ? [As2: list_a,Bs2: list_a] :
( ( ( append_a @ As2 @ ( cons_a @ Y @ Bs2 ) )
= Xs )
& ( ( ldeep_s_a @ Sel @ Xs @ Y )
= ( list_sel_aux_a @ Sel @ Bs2 @ Y ) ) ) ) ).
% ldeep_s_eq_list_sel_aux'_split
thf(fact_405_ldeep__s_Oelims,axiom,
! [X3: a > a > real,Xa: list_a,Y: a > real] :
( ( ( ldeep_s_a @ X3 @ Xa )
= Y )
=> ( ( ( Xa = nil_a )
=> ( Y
!= ( ^ [Uu3: a] : one_one_real ) ) )
=> ~ ! [X2: a,Xs3: list_a] :
( ( Xa
= ( cons_a @ X2 @ Xs3 ) )
=> ( Y
!= ( ^ [A2: a] : ( if_real @ ( A2 = X2 ) @ ( list_sel_aux_a @ X3 @ Xs3 @ A2 ) @ ( ldeep_s_a @ X3 @ Xs3 @ A2 ) ) ) ) ) ) ) ).
% ldeep_s.elims
thf(fact_406_create__ldeep__rev_Osimps_I2_J,axiom,
! [X3: a] :
( ( create_ldeep_rev_a @ ( cons_a @ X3 @ nil_a ) )
= ( relation_a @ X3 ) ) ).
% create_ldeep_rev.simps(2)
thf(fact_407_create__ldeep__rev_Osimps_I3_J,axiom,
! [X3: a,V2: a,Va2: list_a] :
( ( create_ldeep_rev_a @ ( cons_a @ X3 @ ( cons_a @ V2 @ Va2 ) ) )
= ( join_a @ ( create_ldeep_rev_a @ ( cons_a @ V2 @ Va2 ) ) @ ( relation_a @ X3 ) ) ) ).
% create_ldeep_rev.simps(3)
thf(fact_408_nonzero__divide__mult__cancel__right,axiom,
! [B: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ B @ ( times_times_complex @ A @ B ) )
= ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_409_nonzero__divide__mult__cancel__right,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ B @ ( times_times_real @ A @ B ) )
= ( divide_divide_real @ one_one_real @ A ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_410_nonzero__divide__mult__cancel__left,axiom,
! [A: complex,B: complex] :
( ( A != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ ( times_times_complex @ A @ B ) )
= ( divide1717551699836669952omplex @ one_one_complex @ B ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_411_nonzero__divide__mult__cancel__left,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ ( times_times_real @ A @ B ) )
= ( divide_divide_real @ one_one_real @ B ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_412_div__mult__self1,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self1
thf(fact_413_div__mult__self2,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self2
thf(fact_414_div__mult__self3,axiom,
! [B: nat,C: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self3
thf(fact_415_div__mult__self4,axiom,
! [B: nat,C: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self4
thf(fact_416_division__ring__divide__zero,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
= zero_zero_complex ) ).
% division_ring_divide_zero
thf(fact_417_division__ring__divide__zero,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% division_ring_divide_zero
thf(fact_418_divide__cancel__right,axiom,
! [A: complex,C: complex,B: complex] :
( ( ( divide1717551699836669952omplex @ A @ C )
= ( divide1717551699836669952omplex @ B @ C ) )
= ( ( C = zero_zero_complex )
| ( A = B ) ) ) ).
% divide_cancel_right
thf(fact_419_divide__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ( divide_divide_real @ A @ C )
= ( divide_divide_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% divide_cancel_right
thf(fact_420_divide__cancel__left,axiom,
! [C: complex,A: complex,B: complex] :
( ( ( divide1717551699836669952omplex @ C @ A )
= ( divide1717551699836669952omplex @ C @ B ) )
= ( ( C = zero_zero_complex )
| ( A = B ) ) ) ).
% divide_cancel_left
thf(fact_421_divide__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ( divide_divide_real @ C @ A )
= ( divide_divide_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% divide_cancel_left
thf(fact_422_divide__eq__0__iff,axiom,
! [A: complex,B: complex] :
( ( ( divide1717551699836669952omplex @ A @ B )
= zero_zero_complex )
= ( ( A = zero_zero_complex )
| ( B = zero_zero_complex ) ) ) ).
% divide_eq_0_iff
thf(fact_423_divide__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divide_eq_0_iff
thf(fact_424_times__divide__eq__left,axiom,
! [B: real,C: real,A: real] :
( ( times_times_real @ ( divide_divide_real @ B @ C ) @ A )
= ( divide_divide_real @ ( times_times_real @ B @ A ) @ C ) ) ).
% times_divide_eq_left
thf(fact_425_divide__divide__eq__left,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
= ( divide_divide_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% divide_divide_eq_left
thf(fact_426_divide__divide__eq__right,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ A @ ( divide_divide_real @ B @ C ) )
= ( divide_divide_real @ ( times_times_real @ A @ C ) @ B ) ) ).
% divide_divide_eq_right
thf(fact_427_times__divide__eq__right,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( divide_divide_real @ B @ C ) )
= ( divide_divide_real @ ( times_times_real @ A @ B ) @ C ) ) ).
% times_divide_eq_right
thf(fact_428_div__mult__mult1__if,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( C = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= zero_zero_nat ) )
& ( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_429_div__mult__mult2,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_430_div__mult__mult1,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_431_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ C @ B ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_432_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_433_nonzero__mult__divide__mult__cancel__right,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_434_nonzero__mult__divide__mult__cancel__right,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_435_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ B @ C ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_436_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ B @ C ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_437_nonzero__mult__divide__mult__cancel__left,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_438_nonzero__mult__divide__mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_439_mult__divide__mult__cancel__left__if,axiom,
! [C: complex,A: complex,B: complex] :
( ( ( C = zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
= zero_zero_complex ) )
& ( ( C != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_440_mult__divide__mult__cancel__left__if,axiom,
! [C: real,A: real,B: real] :
( ( ( C = zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= zero_zero_real ) )
& ( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_441_divide__eq__1__iff,axiom,
! [A: complex,B: complex] :
( ( ( divide1717551699836669952omplex @ A @ B )
= one_one_complex )
= ( ( B != zero_zero_complex )
& ( A = B ) ) ) ).
% divide_eq_1_iff
thf(fact_442_divide__eq__1__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= one_one_real )
= ( ( B != zero_zero_real )
& ( A = B ) ) ) ).
% divide_eq_1_iff
thf(fact_443_one__eq__divide__iff,axiom,
! [A: complex,B: complex] :
( ( one_one_complex
= ( divide1717551699836669952omplex @ A @ B ) )
= ( ( B != zero_zero_complex )
& ( A = B ) ) ) ).
% one_eq_divide_iff
thf(fact_444_one__eq__divide__iff,axiom,
! [A: real,B: real] :
( ( one_one_real
= ( divide_divide_real @ A @ B ) )
= ( ( B != zero_zero_real )
& ( A = B ) ) ) ).
% one_eq_divide_iff
thf(fact_445_divide__self,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ A )
= one_one_complex ) ) ).
% divide_self
thf(fact_446_divide__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% divide_self
thf(fact_447_divide__self__if,axiom,
! [A: complex] :
( ( ( A = zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ A )
= zero_zero_complex ) )
& ( ( A != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ A )
= one_one_complex ) ) ) ).
% divide_self_if
thf(fact_448_divide__self__if,axiom,
! [A: real] :
( ( ( A = zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= zero_zero_real ) )
& ( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ) ).
% divide_self_if
thf(fact_449_divide__eq__eq__1,axiom,
! [B: real,A: real] :
( ( ( divide_divide_real @ B @ A )
= one_one_real )
= ( ( A != zero_zero_real )
& ( A = B ) ) ) ).
% divide_eq_eq_1
thf(fact_450_eq__divide__eq__1,axiom,
! [B: real,A: real] :
( ( one_one_real
= ( divide_divide_real @ B @ A ) )
= ( ( A != zero_zero_real )
& ( A = B ) ) ) ).
% eq_divide_eq_1
thf(fact_451_one__divide__eq__0__iff,axiom,
! [A: real] :
( ( ( divide_divide_real @ one_one_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% one_divide_eq_0_iff
thf(fact_452_zero__eq__1__divide__iff,axiom,
! [A: real] :
( ( zero_zero_real
= ( divide_divide_real @ one_one_real @ A ) )
= ( A = zero_zero_real ) ) ).
% zero_eq_1_divide_iff
thf(fact_453_times__divide__times__eq,axiom,
! [X3: real,Y: real,Z: real,W: real] :
( ( times_times_real @ ( divide_divide_real @ X3 @ Y ) @ ( divide_divide_real @ Z @ W ) )
= ( divide_divide_real @ ( times_times_real @ X3 @ Z ) @ ( times_times_real @ Y @ W ) ) ) ).
% times_divide_times_eq
thf(fact_454_divide__divide__times__eq,axiom,
! [X3: real,Y: real,Z: real,W: real] :
( ( divide_divide_real @ ( divide_divide_real @ X3 @ Y ) @ ( divide_divide_real @ Z @ W ) )
= ( divide_divide_real @ ( times_times_real @ X3 @ W ) @ ( times_times_real @ Y @ Z ) ) ) ).
% divide_divide_times_eq
thf(fact_455_divide__divide__eq__left_H,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
= ( divide_divide_real @ A @ ( times_times_real @ C @ B ) ) ) ).
% divide_divide_eq_left'
thf(fact_456_add__divide__distrib,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% add_divide_distrib
thf(fact_457_nonzero__eq__divide__eq,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( A
= ( divide1717551699836669952omplex @ B @ C ) )
= ( ( times_times_complex @ A @ C )
= B ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_458_nonzero__eq__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( A
= ( divide_divide_real @ B @ C ) )
= ( ( times_times_real @ A @ C )
= B ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_459_nonzero__divide__eq__eq,axiom,
! [C: complex,B: complex,A: complex] :
( ( C != zero_zero_complex )
=> ( ( ( divide1717551699836669952omplex @ B @ C )
= A )
= ( B
= ( times_times_complex @ A @ C ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_460_nonzero__divide__eq__eq,axiom,
! [C: real,B: real,A: real] :
( ( C != zero_zero_real )
=> ( ( ( divide_divide_real @ B @ C )
= A )
= ( B
= ( times_times_real @ A @ C ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_461_eq__divide__imp,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( ( times_times_complex @ A @ C )
= B )
=> ( A
= ( divide1717551699836669952omplex @ B @ C ) ) ) ) ).
% eq_divide_imp
thf(fact_462_eq__divide__imp,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A @ C )
= B )
=> ( A
= ( divide_divide_real @ B @ C ) ) ) ) ).
% eq_divide_imp
thf(fact_463_divide__eq__imp,axiom,
! [C: complex,B: complex,A: complex] :
( ( C != zero_zero_complex )
=> ( ( B
= ( times_times_complex @ A @ C ) )
=> ( ( divide1717551699836669952omplex @ B @ C )
= A ) ) ) ).
% divide_eq_imp
thf(fact_464_divide__eq__imp,axiom,
! [C: real,B: real,A: real] :
( ( C != zero_zero_real )
=> ( ( B
= ( times_times_real @ A @ C ) )
=> ( ( divide_divide_real @ B @ C )
= A ) ) ) ).
% divide_eq_imp
thf(fact_465_eq__divide__eq,axiom,
! [A: complex,B: complex,C: complex] :
( ( A
= ( divide1717551699836669952omplex @ B @ C ) )
= ( ( ( C != zero_zero_complex )
=> ( ( times_times_complex @ A @ C )
= B ) )
& ( ( C = zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% eq_divide_eq
thf(fact_466_eq__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( A
= ( divide_divide_real @ B @ C ) )
= ( ( ( C != zero_zero_real )
=> ( ( times_times_real @ A @ C )
= B ) )
& ( ( C = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_divide_eq
thf(fact_467_divide__eq__eq,axiom,
! [B: complex,C: complex,A: complex] :
( ( ( divide1717551699836669952omplex @ B @ C )
= A )
= ( ( ( C != zero_zero_complex )
=> ( B
= ( times_times_complex @ A @ C ) ) )
& ( ( C = zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% divide_eq_eq
thf(fact_468_divide__eq__eq,axiom,
! [B: real,C: real,A: real] :
( ( ( divide_divide_real @ B @ C )
= A )
= ( ( ( C != zero_zero_real )
=> ( B
= ( times_times_real @ A @ C ) ) )
& ( ( C = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% divide_eq_eq
thf(fact_469_frac__eq__eq,axiom,
! [Y: complex,Z: complex,X3: complex,W: complex] :
( ( Y != zero_zero_complex )
=> ( ( Z != zero_zero_complex )
=> ( ( ( divide1717551699836669952omplex @ X3 @ Y )
= ( divide1717551699836669952omplex @ W @ Z ) )
= ( ( times_times_complex @ X3 @ Z )
= ( times_times_complex @ W @ Y ) ) ) ) ) ).
% frac_eq_eq
thf(fact_470_frac__eq__eq,axiom,
! [Y: real,Z: real,X3: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( ( divide_divide_real @ X3 @ Y )
= ( divide_divide_real @ W @ Z ) )
= ( ( times_times_real @ X3 @ Z )
= ( times_times_real @ W @ Y ) ) ) ) ) ).
% frac_eq_eq
thf(fact_471_right__inverse__eq,axiom,
! [B: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( ( ( divide1717551699836669952omplex @ A @ B )
= one_one_complex )
= ( A = B ) ) ) ).
% right_inverse_eq
thf(fact_472_right__inverse__eq,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( ( divide_divide_real @ A @ B )
= one_one_real )
= ( A = B ) ) ) ).
% right_inverse_eq
thf(fact_473_divide__add__eq__iff,axiom,
! [Z: complex,X3: complex,Y: complex] :
( ( Z != zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X3 @ Z ) @ Y )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ X3 @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).
% divide_add_eq_iff
thf(fact_474_divide__add__eq__iff,axiom,
! [Z: real,X3: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X3 @ Z ) @ Y )
= ( divide_divide_real @ ( plus_plus_real @ X3 @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% divide_add_eq_iff
thf(fact_475_add__divide__eq__iff,axiom,
! [Z: complex,X3: complex,Y: complex] :
( ( Z != zero_zero_complex )
=> ( ( plus_plus_complex @ X3 @ ( divide1717551699836669952omplex @ Y @ Z ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X3 @ Z ) @ Y ) @ Z ) ) ) ).
% add_divide_eq_iff
thf(fact_476_add__divide__eq__iff,axiom,
! [Z: real,X3: real,Y: real] :
( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ X3 @ ( divide_divide_real @ Y @ Z ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X3 @ Z ) @ Y ) @ Z ) ) ) ).
% add_divide_eq_iff
thf(fact_477_add__num__frac,axiom,
! [Y: complex,Z: complex,X3: complex] :
( ( Y != zero_zero_complex )
=> ( ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ X3 @ Y ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ X3 @ ( times_times_complex @ Z @ Y ) ) @ Y ) ) ) ).
% add_num_frac
thf(fact_478_add__num__frac,axiom,
! [Y: real,Z: real,X3: real] :
( ( Y != zero_zero_real )
=> ( ( plus_plus_real @ Z @ ( divide_divide_real @ X3 @ Y ) )
= ( divide_divide_real @ ( plus_plus_real @ X3 @ ( times_times_real @ Z @ Y ) ) @ Y ) ) ) ).
% add_num_frac
thf(fact_479_add__frac__num,axiom,
! [Y: complex,X3: complex,Z: complex] :
( ( Y != zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X3 @ Y ) @ Z )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ X3 @ ( times_times_complex @ Z @ Y ) ) @ Y ) ) ) ).
% add_frac_num
thf(fact_480_add__frac__num,axiom,
! [Y: real,X3: real,Z: real] :
( ( Y != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X3 @ Y ) @ Z )
= ( divide_divide_real @ ( plus_plus_real @ X3 @ ( times_times_real @ Z @ Y ) ) @ Y ) ) ) ).
% add_frac_num
thf(fact_481_add__frac__eq,axiom,
! [Y: complex,Z: complex,X3: complex,W: complex] :
( ( Y != zero_zero_complex )
=> ( ( Z != zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X3 @ Y ) @ ( divide1717551699836669952omplex @ W @ Z ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X3 @ Z ) @ ( times_times_complex @ W @ Y ) ) @ ( times_times_complex @ Y @ Z ) ) ) ) ) ).
% add_frac_eq
thf(fact_482_add__frac__eq,axiom,
! [Y: real,Z: real,X3: real,W: real] :
( ( Y != zero_zero_real )
=> ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X3 @ Y ) @ ( divide_divide_real @ W @ Z ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X3 @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) ) ) ) ).
% add_frac_eq
thf(fact_483_add__divide__eq__if__simps_I1_J,axiom,
! [Z: complex,A: complex,B: complex] :
( ( ( Z = zero_zero_complex )
=> ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
= A ) )
& ( ( Z != zero_zero_complex )
=> ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_484_add__divide__eq__if__simps_I1_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
= A ) )
& ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_485_add__divide__eq__if__simps_I2_J,axiom,
! [Z: complex,A: complex,B: complex] :
( ( ( Z = zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
= B ) )
& ( ( Z != zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_486_add__divide__eq__if__simps_I2_J,axiom,
! [Z: real,A: real,B: real] :
( ( ( Z = zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
= B ) )
& ( ( Z != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
= ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_487_div__add__self1,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self1
thf(fact_488_div__add__self2,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self2
thf(fact_489_real__divide__square__eq,axiom,
! [R2: real,A: real] :
( ( divide_divide_real @ ( times_times_real @ R2 @ A ) @ ( times_times_real @ R2 @ R2 ) )
= ( divide_divide_real @ A @ R2 ) ) ).
% real_divide_square_eq
thf(fact_490_bits__div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% bits_div_by_0
thf(fact_491_bits__div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_div_0
thf(fact_492_dbl__inc__simps_I2_J,axiom,
( ( neg_nu8295874005876285629c_real @ zero_zero_real )
= one_one_real ) ).
% dbl_inc_simps(2)
thf(fact_493_dbl__inc__simps_I2_J,axiom,
( ( neg_nu8557863876264182079omplex @ zero_zero_complex )
= one_one_complex ) ).
% dbl_inc_simps(2)
thf(fact_494_dbl__inc__def,axiom,
( neg_nu8557863876264182079omplex
= ( ^ [X4: complex] : ( plus_plus_complex @ ( plus_plus_complex @ X4 @ X4 ) @ one_one_complex ) ) ) ).
% dbl_inc_def
thf(fact_495_dbl__inc__def,axiom,
( neg_nu8295874005876285629c_real
= ( ^ [X4: real] : ( plus_plus_real @ ( plus_plus_real @ X4 @ X4 ) @ one_one_real ) ) ) ).
% dbl_inc_def
thf(fact_496_create__rdeep_Opelims,axiom,
! [X3: list_a,Y: joinTree_a] :
( ( ( create_rdeep_a @ X3 )
= Y )
=> ( ( accp_list_a @ create_rdeep_rel_a @ X3 )
=> ( ( ( X3 = nil_a )
=> ( ( Y = undefined_joinTree_a )
=> ~ ( accp_list_a @ create_rdeep_rel_a @ nil_a ) ) )
=> ( ! [X2: a] :
( ( X3
= ( cons_a @ X2 @ nil_a ) )
=> ( ( Y
= ( relation_a @ X2 ) )
=> ~ ( accp_list_a @ create_rdeep_rel_a @ ( cons_a @ X2 @ nil_a ) ) ) )
=> ~ ! [X2: a,V: a,Va: list_a] :
( ( X3
= ( cons_a @ X2 @ ( cons_a @ V @ Va ) ) )
=> ( ( Y
= ( join_a @ ( relation_a @ X2 ) @ ( create_rdeep_a @ ( cons_a @ V @ Va ) ) ) )
=> ~ ( accp_list_a @ create_rdeep_rel_a @ ( cons_a @ X2 @ ( cons_a @ V @ Va ) ) ) ) ) ) ) ) ) ).
% create_rdeep.pelims
thf(fact_497_create__ldeep__rev_Opelims,axiom,
! [X3: list_a,Y: joinTree_a] :
( ( ( create_ldeep_rev_a @ X3 )
= Y )
=> ( ( accp_list_a @ create4822648437634936863_rel_a @ X3 )
=> ( ( ( X3 = nil_a )
=> ( ( Y = undefined_joinTree_a )
=> ~ ( accp_list_a @ create4822648437634936863_rel_a @ nil_a ) ) )
=> ( ! [X2: a] :
( ( X3
= ( cons_a @ X2 @ nil_a ) )
=> ( ( Y
= ( relation_a @ X2 ) )
=> ~ ( accp_list_a @ create4822648437634936863_rel_a @ ( cons_a @ X2 @ nil_a ) ) ) )
=> ~ ! [X2: a,V: a,Va: list_a] :
( ( X3
= ( cons_a @ X2 @ ( cons_a @ V @ Va ) ) )
=> ( ( Y
= ( join_a @ ( create_ldeep_rev_a @ ( cons_a @ V @ Va ) ) @ ( relation_a @ X2 ) ) )
=> ~ ( accp_list_a @ create4822648437634936863_rel_a @ ( cons_a @ X2 @ ( cons_a @ V @ Va ) ) ) ) ) ) ) ) ) ).
% create_ldeep_rev.pelims
thf(fact_498_arcosh__1,axiom,
( ( arcosh_real @ one_one_real )
= zero_zero_real ) ).
% arcosh_1
thf(fact_499_artanh__0,axiom,
( ( artanh_real @ zero_zero_real )
= zero_zero_real ) ).
% artanh_0
thf(fact_500_arsinh__0,axiom,
( ( arsinh_real @ zero_zero_real )
= zero_zero_real ) ).
% arsinh_0
thf(fact_501_ln__one,axiom,
( ( ln_ln_real @ one_one_real )
= zero_zero_real ) ).
% ln_one
thf(fact_502_prod__list_Oeq__foldr,axiom,
( groups6371653412389394274st_nat
= ( ^ [Xs2: list_nat] : ( foldr_nat_nat @ times_times_nat @ Xs2 @ one_one_nat ) ) ) ).
% prod_list.eq_foldr
thf(fact_503_prod__list_Oeq__foldr,axiom,
( groups7979759902575632448omplex
= ( ^ [Xs2: list_complex] : ( foldr_2100044862368902405omplex @ times_times_complex @ Xs2 @ one_one_complex ) ) ) ).
% prod_list.eq_foldr
thf(fact_504_prod__list_Oeq__foldr,axiom,
( groups2776710990603637054t_real
= ( ^ [Xs2: list_real] : ( foldr_real_real @ times_times_real @ Xs2 @ one_one_real ) ) ) ).
% prod_list.eq_foldr
thf(fact_505_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_506_add__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_507_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_508_add__le__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_509_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_510_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_511_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_512_le__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel2
thf(fact_513_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_514_le__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel1
thf(fact_515_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_516_add__le__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel2
thf(fact_517_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_518_add__le__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel1
thf(fact_519_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_520_prod__list_OCons,axiom,
! [X3: real,Xs: list_real] :
( ( groups2776710990603637054t_real @ ( cons_real @ X3 @ Xs ) )
= ( times_times_real @ X3 @ ( groups2776710990603637054t_real @ Xs ) ) ) ).
% prod_list.Cons
thf(fact_521_prod__list_Oappend,axiom,
! [Xs: list_real,Ys: list_real] :
( ( groups2776710990603637054t_real @ ( append_real @ Xs @ Ys ) )
= ( times_times_real @ ( groups2776710990603637054t_real @ Xs ) @ ( groups2776710990603637054t_real @ Ys ) ) ) ).
% prod_list.append
thf(fact_522_zero__le__divide__1__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% zero_le_divide_1_iff
thf(fact_523_divide__le__0__1__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% divide_le_0_1_iff
thf(fact_524_verit__comp__simplify1_I2_J,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_525_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_526_verit__la__disequality,axiom,
! [A: real,B: real] :
( ( A = B )
| ~ ( ord_less_eq_real @ A @ B )
| ~ ( ord_less_eq_real @ B @ A ) ) ).
% verit_la_disequality
thf(fact_527_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_528_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_529_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_530_add__le__imp__le__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_531_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_532_add__le__imp__le__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_533_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_534_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
? [C2: nat] :
( B2
= ( plus_plus_nat @ A2 @ C2 ) ) ) ) ).
% le_iff_add
thf(fact_535_add__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_right_mono
thf(fact_536_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_537_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C3: nat] :
( B
!= ( plus_plus_nat @ A @ C3 ) ) ) ).
% less_eqE
thf(fact_538_add__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_left_mono
thf(fact_539_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_540_add__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_mono
thf(fact_541_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_542_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_543_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_544_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_545_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_546_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_547_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_548_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_549_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_550_zero__le,axiom,
! [X3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X3 ) ).
% zero_le
thf(fact_551_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_552_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_553_zero__le__mult__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% zero_le_mult_iff
thf(fact_554_mult__nonneg__nonpos2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_555_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_556_mult__nonpos__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_nonpos_nonneg
thf(fact_557_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_558_mult__nonneg__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos
thf(fact_559_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_560_mult__nonneg__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_561_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_562_split__mult__neg__le,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ).
% split_mult_neg_le
thf(fact_563_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_564_mult__le__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_565_mult__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_566_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_567_mult__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_568_mult__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_569_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_570_mult__nonpos__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_571_mult__left__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_572_split__mult__pos__le,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_573_zero__le__square,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% zero_le_square
thf(fact_574_mult__mono_H,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_575_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_576_mult__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_577_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_578_add__nonpos__eq__0__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ( ( plus_plus_real @ X3 @ Y )
= zero_zero_real )
= ( ( X3 = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_579_add__nonpos__eq__0__iff,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X3 @ Y )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_580_add__nonneg__eq__0__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ( plus_plus_real @ X3 @ Y )
= zero_zero_real )
= ( ( X3 = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_581_add__nonneg__eq__0__iff,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X3 @ Y )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_582_add__nonpos__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_nonpos
thf(fact_583_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_584_add__nonneg__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_585_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_586_add__increasing2,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_587_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_588_add__decreasing2,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_589_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_590_add__increasing,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_591_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_592_add__decreasing,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_593_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_594_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_le_one
thf(fact_595_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_596_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_597_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_598_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_599_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_600_divide__right__mono__neg,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).
% divide_right_mono_neg
thf(fact_601_divide__nonpos__nonpos,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_602_divide__nonpos__nonneg,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonpos_nonneg
thf(fact_603_divide__nonneg__nonpos,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonneg_nonpos
thf(fact_604_divide__nonneg__nonneg,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_605_zero__le__divide__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% zero_le_divide_iff
thf(fact_606_divide__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_right_mono
thf(fact_607_divide__le__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% divide_le_0_iff
thf(fact_608_ln__ge__zero,axiom,
! [X3: real] :
( ( ord_less_eq_real @ one_one_real @ X3 )
=> ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X3 ) ) ) ).
% ln_ge_zero
thf(fact_609_prod__list__zero__iff,axiom,
! [Xs: list_real] :
( ( ( groups2776710990603637054t_real @ Xs )
= zero_zero_real )
= ( member_real @ zero_zero_real @ ( set_real2 @ Xs ) ) ) ).
% prod_list_zero_iff
thf(fact_610_prod__list__zero__iff,axiom,
! [Xs: list_nat] :
( ( ( groups6371653412389394274st_nat @ Xs )
= zero_zero_nat )
= ( member_nat @ zero_zero_nat @ ( set_nat2 @ Xs ) ) ) ).
% prod_list_zero_iff
thf(fact_611_prod__list__zero__iff,axiom,
! [Xs: list_complex] :
( ( ( groups7979759902575632448omplex @ Xs )
= zero_zero_complex )
= ( member_complex @ zero_zero_complex @ ( set_complex2 @ Xs ) ) ) ).
% prod_list_zero_iff
thf(fact_612_sum__squares__le__zero__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real )
= ( ( X3 = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_613_sum__squares__ge__zero,axiom,
! [X3: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_614_mult__left__le__one__le,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Y @ X3 ) @ X3 ) ) ) ) ).
% mult_left_le_one_le
thf(fact_615_mult__right__le__one__le,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ X3 @ Y ) @ X3 ) ) ) ) ).
% mult_right_le_one_le
thf(fact_616_mult__le__one,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ B @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ one_one_real ) ) ) ) ).
% mult_le_one
thf(fact_617_mult__le__one,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_618_mult__left__le,axiom,
! [C: real,A: real] :
( ( ord_less_eq_real @ C @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_619_mult__left__le,axiom,
! [C: nat,A: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_620_ln__add__one__self__le__self,axiom,
! [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X3 ) ) @ X3 ) ) ).
% ln_add_one_self_le_self
thf(fact_621_convex__bound__le,axiom,
! [X3: real,A: real,Y: real,U: real,V2: real] :
( ( ord_less_eq_real @ X3 @ A )
=> ( ( ord_less_eq_real @ Y @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ U )
=> ( ( ord_less_eq_real @ zero_zero_real @ V2 )
=> ( ( ( plus_plus_real @ U @ V2 )
= one_one_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X3 ) @ ( times_times_real @ V2 @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_622_reasonable__cards_Oelims_I3_J,axiom,
! [X3: a > real,Xa: a > a > real,Xb: joinTree_a] :
( ~ ( reasonable_cards_a @ X3 @ Xa @ Xb )
=> ( ! [Rel2: a] :
( ( Xb
= ( relation_a @ Rel2 ) )
=> ( ord_less_real @ zero_zero_real @ ( X3 @ Rel2 ) ) )
=> ~ ! [L2: joinTree_a,R: joinTree_a] :
( ( Xb
= ( join_a @ L2 @ R ) )
=> ( ( ord_less_eq_real @ ( card_a @ X3 @ Xa @ ( join_a @ L2 @ R ) ) @ ( times_times_real @ ( card_a @ X3 @ Xa @ L2 ) @ ( card_a @ X3 @ Xa @ R ) ) )
& ( ord_less_real @ zero_zero_real @ ( card_a @ X3 @ Xa @ ( join_a @ L2 @ R ) ) )
& ( reasonable_cards_a @ X3 @ Xa @ L2 )
& ( reasonable_cards_a @ X3 @ Xa @ R ) ) ) ) ) ).
% reasonable_cards.elims(3)
thf(fact_623_reasonable__cards_Oelims_I2_J,axiom,
! [X3: a > real,Xa: a > a > real,Xb: joinTree_a] :
( ( reasonable_cards_a @ X3 @ Xa @ Xb )
=> ( ! [Rel2: a] :
( ( Xb
= ( relation_a @ Rel2 ) )
=> ~ ( ord_less_real @ zero_zero_real @ ( X3 @ Rel2 ) ) )
=> ~ ! [L2: joinTree_a,R: joinTree_a] :
( ( Xb
= ( join_a @ L2 @ R ) )
=> ~ ( ( ord_less_eq_real @ ( card_a @ X3 @ Xa @ ( join_a @ L2 @ R ) ) @ ( times_times_real @ ( card_a @ X3 @ Xa @ L2 ) @ ( card_a @ X3 @ Xa @ R ) ) )
& ( ord_less_real @ zero_zero_real @ ( card_a @ X3 @ Xa @ ( join_a @ L2 @ R ) ) )
& ( reasonable_cards_a @ X3 @ Xa @ L2 )
& ( reasonable_cards_a @ X3 @ Xa @ R ) ) ) ) ) ).
% reasonable_cards.elims(2)
thf(fact_624_reasonable__cards_Oelims_I1_J,axiom,
! [X3: a > real,Xa: a > a > real,Xb: joinTree_a,Y: $o] :
( ( ( reasonable_cards_a @ X3 @ Xa @ Xb )
= Y )
=> ( ! [Rel2: a] :
( ( Xb
= ( relation_a @ Rel2 ) )
=> ( Y
= ( ~ ( ord_less_real @ zero_zero_real @ ( X3 @ Rel2 ) ) ) ) )
=> ~ ! [L2: joinTree_a,R: joinTree_a] :
( ( Xb
= ( join_a @ L2 @ R ) )
=> ( Y
= ( ~ ( ( ord_less_eq_real @ ( card_a @ X3 @ Xa @ ( join_a @ L2 @ R ) ) @ ( times_times_real @ ( card_a @ X3 @ Xa @ L2 ) @ ( card_a @ X3 @ Xa @ R ) ) )
& ( ord_less_real @ zero_zero_real @ ( card_a @ X3 @ Xa @ ( join_a @ L2 @ R ) ) )
& ( reasonable_cards_a @ X3 @ Xa @ L2 )
& ( reasonable_cards_a @ X3 @ Xa @ R ) ) ) ) ) ) ) ).
% reasonable_cards.elims(1)
thf(fact_625_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_626_add__less__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_627_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_628_add__less__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_629_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_630_add__less__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel1
thf(fact_631_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_632_add__less__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel2
thf(fact_633_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_634_less__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel1
thf(fact_635_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_636_less__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel2
thf(fact_637_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_638_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_639_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_640_not__real__square__gt__zero,axiom,
! [X3: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X3 @ X3 ) ) )
= ( X3 = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_641_ln__inj__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ( ln_ln_real @ X3 )
= ( ln_ln_real @ Y ) )
= ( X3 = Y ) ) ) ) ).
% ln_inj_iff
thf(fact_642_ln__less__cancel__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ Y ) )
= ( ord_less_real @ X3 @ Y ) ) ) ) ).
% ln_less_cancel_iff
thf(fact_643_divide__less__0__1__iff,axiom,
! [A: real] :
( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% divide_less_0_1_iff
thf(fact_644_divide__less__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_real @ A @ B ) ) ) ).
% divide_less_eq_1_neg
thf(fact_645_divide__less__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_real @ B @ A ) ) ) ).
% divide_less_eq_1_pos
thf(fact_646_less__divide__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_real @ B @ A ) ) ) ).
% less_divide_eq_1_neg
thf(fact_647_less__divide__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_real @ A @ B ) ) ) ).
% less_divide_eq_1_pos
thf(fact_648_zero__less__divide__1__iff,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% zero_less_divide_1_iff
thf(fact_649_ln__le__cancel__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ Y ) )
= ( ord_less_eq_real @ X3 @ Y ) ) ) ) ).
% ln_le_cancel_iff
thf(fact_650_ln__eq__zero__iff,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ( ln_ln_real @ X3 )
= zero_zero_real )
= ( X3 = one_one_real ) ) ) ).
% ln_eq_zero_iff
thf(fact_651_ln__gt__zero__iff,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X3 ) )
= ( ord_less_real @ one_one_real @ X3 ) ) ) ).
% ln_gt_zero_iff
thf(fact_652_ln__less__zero__iff,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ ( ln_ln_real @ X3 ) @ zero_zero_real )
= ( ord_less_real @ X3 @ one_one_real ) ) ) ).
% ln_less_zero_iff
thf(fact_653_divide__le__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% divide_le_eq_1_neg
thf(fact_654_divide__le__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_eq_real @ B @ A ) ) ) ).
% divide_le_eq_1_pos
thf(fact_655_le__divide__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_eq_real @ B @ A ) ) ) ).
% le_divide_eq_1_neg
thf(fact_656_le__divide__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% le_divide_eq_1_pos
thf(fact_657_ln__le__zero__iff,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ ( ln_ln_real @ X3 ) @ zero_zero_real )
= ( ord_less_eq_real @ X3 @ one_one_real ) ) ) ).
% ln_le_zero_iff
thf(fact_658_ln__ge__zero__iff,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X3 ) )
= ( ord_less_eq_real @ one_one_real @ X3 ) ) ) ).
% ln_ge_zero_iff
thf(fact_659_verit__comp__simplify1_I3_J,axiom,
! [B4: real,A4: real] :
( ( ~ ( ord_less_eq_real @ B4 @ A4 ) )
= ( ord_less_real @ A4 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_660_verit__comp__simplify1_I3_J,axiom,
! [B4: nat,A4: nat] :
( ( ~ ( ord_less_eq_nat @ B4 @ A4 ) )
= ( ord_less_nat @ A4 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_661_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_662_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_663_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_664_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_665_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_666_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_667_add__mono__thms__linordered__field_I5_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_668_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_669_add__mono__thms__linordered__field_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_670_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_671_add__mono__thms__linordered__field_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( K = L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_672_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_673_add__strict__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_674_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_675_add__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_676_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_677_add__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_678_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_679_add__less__imp__less__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_680_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_681_add__less__imp__less__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_682_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_683_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_684_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_685_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_686_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_687_linorder__neqE__linordered__idom,axiom,
! [X3: real,Y: real] :
( ( X3 != Y )
=> ( ~ ( ord_less_real @ X3 @ Y )
=> ( ord_less_real @ Y @ X3 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_688_ln__less__self,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ord_less_real @ ( ln_ln_real @ X3 ) @ X3 ) ) ).
% ln_less_self
thf(fact_689_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_690_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_691_mult__less__cancel__right__disj,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_692_mult__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_693_mult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_694_mult__strict__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_695_mult__less__cancel__left__disj,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_696_mult__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_697_mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_698_mult__strict__left__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_699_mult__less__cancel__left__pos,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_real @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_700_mult__less__cancel__left__neg,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_real @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_701_zero__less__mult__pos2,axiom,
! [B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_702_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_703_zero__less__mult__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_704_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_705_zero__less__mult__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% zero_less_mult_iff
thf(fact_706_mult__pos__neg2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% mult_pos_neg2
thf(fact_707_mult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_708_mult__pos__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_709_mult__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_710_mult__pos__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_pos_neg
thf(fact_711_mult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_712_mult__neg__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_neg_pos
thf(fact_713_mult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_714_mult__less__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_715_not__square__less__zero,axiom,
! [A: real] :
~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_716_mult__neg__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_717_add__less__le__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_718_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_719_add__le__less__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_720_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_721_add__mono__thms__linordered__field_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_722_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_723_add__mono__thms__linordered__field_I4_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_724_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_725_add__less__zeroD,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ ( plus_plus_real @ X3 @ Y ) @ zero_zero_real )
=> ( ( ord_less_real @ X3 @ zero_zero_real )
| ( ord_less_real @ Y @ zero_zero_real ) ) ) ).
% add_less_zeroD
thf(fact_726_pos__add__strict,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_727_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_728_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C3: nat] :
( ( B
= ( plus_plus_nat @ A @ C3 ) )
=> ( C3 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_729_add__pos__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_730_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_731_add__neg__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_neg_neg
thf(fact_732_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_733_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_734_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_735_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one
thf(fact_736_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_737_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_738_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_739_divide__neg__neg,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ zero_zero_real )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y ) ) ) ) ).
% divide_neg_neg
thf(fact_740_divide__neg__pos,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ X3 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_real @ ( divide_divide_real @ X3 @ Y ) @ zero_zero_real ) ) ) ).
% divide_neg_pos
thf(fact_741_divide__pos__neg,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ X3 @ Y ) @ zero_zero_real ) ) ) ).
% divide_pos_neg
thf(fact_742_divide__pos__pos,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y ) ) ) ) ).
% divide_pos_pos
thf(fact_743_divide__less__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% divide_less_0_iff
thf(fact_744_divide__less__cancel,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ A ) )
& ( C != zero_zero_real ) ) ) ).
% divide_less_cancel
thf(fact_745_zero__less__divide__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% zero_less_divide_iff
thf(fact_746_divide__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_strict_right_mono
thf(fact_747_divide__strict__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_748_less__1__mult,axiom,
! [M: real,N: real] :
( ( ord_less_real @ one_one_real @ M )
=> ( ( ord_less_real @ one_one_real @ N )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_749_less__1__mult,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_750_less__add__one,axiom,
! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).
% less_add_one
thf(fact_751_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_752_add__mono1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B @ one_one_real ) ) ) ).
% add_mono1
thf(fact_753_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_754_ln__bound,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ord_less_eq_real @ ( ln_ln_real @ X3 ) @ X3 ) ) ).
% ln_bound
thf(fact_755_ln__gt__zero,axiom,
! [X3: real] :
( ( ord_less_real @ one_one_real @ X3 )
=> ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X3 ) ) ) ).
% ln_gt_zero
thf(fact_756_ln__less__zero,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ X3 @ one_one_real )
=> ( ord_less_real @ ( ln_ln_real @ X3 ) @ zero_zero_real ) ) ) ).
% ln_less_zero
thf(fact_757_ln__gt__zero__imp__gt__one,axiom,
! [X3: real] :
( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X3 ) )
=> ( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ord_less_real @ one_one_real @ X3 ) ) ) ).
% ln_gt_zero_imp_gt_one
thf(fact_758_mult__less__le__imp__less,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_759_mult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_760_mult__le__less__imp__less,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_761_mult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_762_mult__right__le__imp__le,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_763_mult__right__le__imp__le,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_764_mult__left__le__imp__le,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_765_mult__left__le__imp__le,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_766_mult__le__cancel__left__pos,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_767_mult__le__cancel__left__neg,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_768_mult__less__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_769_mult__strict__mono_H,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_770_mult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_771_mult__right__less__imp__less,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_772_mult__right__less__imp__less,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_773_mult__less__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_774_mult__strict__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_775_mult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_776_mult__left__less__imp__less,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_777_mult__left__less__imp__less,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_778_mult__le__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_779_mult__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_780_field__le__epsilon,axiom,
! [X3: real,Y: real] :
( ! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
=> ( ord_less_eq_real @ X3 @ ( plus_plus_real @ Y @ E2 ) ) )
=> ( ord_less_eq_real @ X3 @ Y ) ) ).
% field_le_epsilon
thf(fact_781_add__strict__increasing2,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_782_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_783_add__strict__increasing,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_784_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_785_add__pos__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_786_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_787_add__nonpos__neg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_neg
thf(fact_788_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_789_add__nonneg__pos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_790_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_791_add__neg__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_neg_nonpos
thf(fact_792_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_793_divide__nonpos__pos,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonpos_pos
thf(fact_794_divide__nonpos__neg,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ X3 @ zero_zero_real )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y ) ) ) ) ).
% divide_nonpos_neg
thf(fact_795_divide__nonneg__pos,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y ) ) ) ) ).
% divide_nonneg_pos
thf(fact_796_divide__nonneg__neg,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonneg_neg
thf(fact_797_divide__le__cancel,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% divide_le_cancel
thf(fact_798_frac__less2,axiom,
! [X3: real,Y: real,W: real,Z: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ X3 @ Y )
=> ( ( ord_less_real @ zero_zero_real @ W )
=> ( ( ord_less_real @ W @ Z )
=> ( ord_less_real @ ( divide_divide_real @ X3 @ Z ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).
% frac_less2
thf(fact_799_frac__less,axiom,
! [X3: real,Y: real,W: real,Z: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ X3 @ Y )
=> ( ( ord_less_real @ zero_zero_real @ W )
=> ( ( ord_less_eq_real @ W @ Z )
=> ( ord_less_real @ ( divide_divide_real @ X3 @ Z ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).
% frac_less
thf(fact_800_frac__le,axiom,
! [Y: real,X3: real,W: real,Z: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ X3 @ Y )
=> ( ( ord_less_real @ zero_zero_real @ W )
=> ( ( ord_less_eq_real @ W @ Z )
=> ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Z ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).
% frac_le
thf(fact_801_not__sum__squares__lt__zero,axiom,
! [X3: real,Y: real] :
~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real ) ).
% not_sum_squares_lt_zero
thf(fact_802_sum__squares__gt__zero__iff,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y @ Y ) ) )
= ( ( X3 != zero_zero_real )
| ( Y != zero_zero_real ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_803_zero__less__two,axiom,
ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% zero_less_two
thf(fact_804_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_805_divide__less__eq,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% divide_less_eq
thf(fact_806_less__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% less_divide_eq
thf(fact_807_neg__divide__less__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% neg_divide_less_eq
thf(fact_808_neg__less__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% neg_less_divide_eq
thf(fact_809_pos__divide__less__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% pos_divide_less_eq
thf(fact_810_pos__less__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% pos_less_divide_eq
thf(fact_811_mult__imp__div__pos__less,axiom,
! [Y: real,X3: real,Z: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ X3 @ ( times_times_real @ Z @ Y ) )
=> ( ord_less_real @ ( divide_divide_real @ X3 @ Y ) @ Z ) ) ) ).
% mult_imp_div_pos_less
thf(fact_812_mult__imp__less__div__pos,axiom,
! [Y: real,Z: real,X3: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( times_times_real @ Z @ Y ) @ X3 )
=> ( ord_less_real @ Z @ ( divide_divide_real @ X3 @ Y ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_813_divide__strict__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% divide_strict_left_mono
thf(fact_814_divide__strict__left__mono__neg,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% divide_strict_left_mono_neg
thf(fact_815_divide__less__eq__1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ A ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ A @ B ) )
| ( A = zero_zero_real ) ) ) ).
% divide_less_eq_1
thf(fact_816_less__divide__eq__1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% less_divide_eq_1
thf(fact_817_less__half__sum,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ A @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).
% less_half_sum
thf(fact_818_gt__half__sum,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B ) ) ).
% gt_half_sum
thf(fact_819_ln__ge__zero__imp__ge__one,axiom,
! [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X3 ) )
=> ( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ord_less_eq_real @ one_one_real @ X3 ) ) ) ).
% ln_ge_zero_imp_ge_one
thf(fact_820_ln__mult,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ln_ln_real @ ( times_times_real @ X3 @ Y ) )
= ( plus_plus_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ Y ) ) ) ) ) ).
% ln_mult
thf(fact_821_reasonable__cards_Osimps_I1_J,axiom,
! [Cf: a > real,F: a > a > real,Rel: a] :
( ( reasonable_cards_a @ Cf @ F @ ( relation_a @ Rel ) )
= ( ord_less_real @ zero_zero_real @ ( Cf @ Rel ) ) ) ).
% reasonable_cards.simps(1)
thf(fact_822_clist__eq__if__cf__eq,axiom,
! [Xs: list_a,Sf: a > real,Cf3: a > real,Cf: a > real,H: a > real,R2: a] :
( ! [X2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ X2 ) @ ( set_a2 @ Xs ) )
=> ( ( ldeep_T_a @ Sf @ Cf3 @ X2 )
= ( ldeep_T_a @ Sf @ Cf @ X2 ) ) )
=> ( ( c_list_a @ Sf @ Cf3 @ H @ R2 @ Xs )
= ( c_list_a @ Sf @ Cf @ H @ R2 @ Xs ) ) ) ).
% clist_eq_if_cf_eq
thf(fact_823_field__le__mult__one__interval,axiom,
! [X3: real,Y: real] :
( ! [Z3: real] :
( ( ord_less_real @ zero_zero_real @ Z3 )
=> ( ( ord_less_real @ Z3 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Z3 @ X3 ) @ Y ) ) )
=> ( ord_less_eq_real @ X3 @ Y ) ) ).
% field_le_mult_one_interval
thf(fact_824_mult__le__cancel__left1,axiom,
! [C: real,B: real] :
( ( ord_less_eq_real @ C @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ one_one_real @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_825_mult__le__cancel__left2,axiom,
! [C: real,A: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ C )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ one_one_real ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_826_mult__le__cancel__right1,axiom,
! [C: real,B: real] :
( ( ord_less_eq_real @ C @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ one_one_real @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_827_mult__le__cancel__right2,axiom,
! [A: real,C: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ C )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ one_one_real ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_828_mult__less__cancel__left1,axiom,
! [C: real,B: real] :
( ( ord_less_real @ C @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ one_one_real @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_829_mult__less__cancel__left2,axiom,
! [C: real,A: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ C )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ one_one_real ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_830_mult__less__cancel__right1,axiom,
! [C: real,B: real] :
( ( ord_less_real @ C @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ one_one_real @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_831_mult__less__cancel__right2,axiom,
! [A: real,C: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ C )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ one_one_real ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_832_divide__left__mono__neg,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% divide_left_mono_neg
thf(fact_833_mult__imp__le__div__pos,axiom,
! [Y: real,Z: real,X3: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ ( times_times_real @ Z @ Y ) @ X3 )
=> ( ord_less_eq_real @ Z @ ( divide_divide_real @ X3 @ Y ) ) ) ) ).
% mult_imp_le_div_pos
thf(fact_834_mult__imp__div__pos__le,axiom,
! [Y: real,X3: real,Z: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ X3 @ ( times_times_real @ Z @ Y ) )
=> ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y ) @ Z ) ) ) ).
% mult_imp_div_pos_le
thf(fact_835_pos__le__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% pos_le_divide_eq
thf(fact_836_pos__divide__le__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% pos_divide_le_eq
thf(fact_837_neg__le__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% neg_le_divide_eq
thf(fact_838_neg__divide__le__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% neg_divide_le_eq
thf(fact_839_divide__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% divide_left_mono
thf(fact_840_le__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% le_divide_eq
thf(fact_841_divide__le__eq,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% divide_le_eq
thf(fact_842_divide__le__eq__1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ A ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ A @ B ) )
| ( A = zero_zero_real ) ) ) ).
% divide_le_eq_1
thf(fact_843_le__divide__eq__1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ A @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ A ) ) ) ) ).
% le_divide_eq_1
thf(fact_844_convex__bound__lt,axiom,
! [X3: real,A: real,Y: real,U: real,V2: real] :
( ( ord_less_real @ X3 @ A )
=> ( ( ord_less_real @ Y @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ U )
=> ( ( ord_less_eq_real @ zero_zero_real @ V2 )
=> ( ( ( plus_plus_real @ U @ V2 )
= one_one_real )
=> ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X3 ) @ ( times_times_real @ V2 @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_845_reasonable__cards_Osimps_I2_J,axiom,
! [Cf: a > real,F: a > a > real,L: joinTree_a,R2: joinTree_a] :
( ( reasonable_cards_a @ Cf @ F @ ( join_a @ L @ R2 ) )
= ( ( ord_less_eq_real @ ( card_a @ Cf @ F @ ( join_a @ L @ R2 ) ) @ ( times_times_real @ ( card_a @ Cf @ F @ L ) @ ( card_a @ Cf @ F @ R2 ) ) )
& ( ord_less_real @ zero_zero_real @ ( card_a @ Cf @ F @ ( join_a @ L @ R2 ) ) )
& ( reasonable_cards_a @ Cf @ F @ L )
& ( reasonable_cards_a @ Cf @ F @ R2 ) ) ) ).
% reasonable_cards.simps(2)
thf(fact_846_mult__le__cancel__iff2,axiom,
! [Z: real,X3: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( ( ord_less_eq_real @ ( times_times_real @ Z @ X3 ) @ ( times_times_real @ Z @ Y ) )
= ( ord_less_eq_real @ X3 @ Y ) ) ) ).
% mult_le_cancel_iff2
thf(fact_847_mult__le__cancel__iff1,axiom,
! [Z: real,X3: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( ( ord_less_eq_real @ ( times_times_real @ X3 @ Z ) @ ( times_times_real @ Y @ Z ) )
= ( ord_less_eq_real @ X3 @ Y ) ) ) ).
% mult_le_cancel_iff1
thf(fact_848_mult__less__iff1,axiom,
! [Z: real,X3: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( ( ord_less_real @ ( times_times_real @ X3 @ Z ) @ ( times_times_real @ Y @ Z ) )
= ( ord_less_real @ X3 @ Y ) ) ) ).
% mult_less_iff1
thf(fact_849_ldeep__s__pos,axiom,
! [F: a > a > real,Xs: list_a,X3: a] :
( ( sel_reasonable_a @ F )
=> ( ord_less_real @ zero_zero_real @ ( ldeep_s_a @ F @ Xs @ X3 ) ) ) ).
% ldeep_s_pos
thf(fact_850_ldeep__s__reasonable,axiom,
! [F: a > a > real,Xs: list_a,Y: a] :
( ( sel_reasonable_a @ F )
=> ( ( ord_less_eq_real @ ( ldeep_s_a @ F @ Xs @ Y ) @ one_one_real )
& ( ord_less_real @ zero_zero_real @ ( ldeep_s_a @ F @ Xs @ Y ) ) ) ) ).
% ldeep_s_reasonable
thf(fact_851_ldeep__T__pos,axiom,
! [Ys: list_a,Cf: a > real,F: a > a > real,Xs: list_a] :
( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Ys ) )
=> ( ord_less_real @ zero_zero_real @ ( Cf @ X2 ) ) )
=> ( ( sel_reasonable_a @ F )
=> ( ord_less_real @ zero_zero_real @ ( ldeep_T_a @ ( ldeep_s_a @ F @ Xs ) @ Cf @ Ys ) ) ) ) ).
% ldeep_T_pos
thf(fact_852_ldeep__T__pos_H,axiom,
! [Xs: list_a,Cf: a > real,F: a > a > real] :
( ( distinct_a @ Xs )
=> ( ( pos_list_cards_a @ Cf @ Xs )
=> ( ( sel_reasonable_a @ F )
=> ( ord_less_real @ zero_zero_real @ ( ldeep_T_a @ ( ldeep_s_a @ F @ Xs ) @ Cf @ Xs ) ) ) ) ) ).
% ldeep_T_pos'
thf(fact_853_set__sel__aux__reasonable__fin,axiom,
! [Y: set_nat,F: nat > nat > real,X3: nat] :
( ( finite_finite_nat @ Y )
=> ( ( sel_reasonable_nat @ F )
=> ( ( ord_less_eq_real @ ( set_sel_aux_nat @ F @ X3 @ Y ) @ one_one_real )
& ( ord_less_real @ zero_zero_real @ ( set_sel_aux_nat @ F @ X3 @ Y ) ) ) ) ) ).
% set_sel_aux_reasonable_fin
thf(fact_854_set__sel__aux__reasonable__fin,axiom,
! [Y: set_complex,F: complex > complex > real,X3: complex] :
( ( finite3207457112153483333omplex @ Y )
=> ( ( sel_re4140149273989857872omplex @ F )
=> ( ( ord_less_eq_real @ ( set_sel_aux_complex @ F @ X3 @ Y ) @ one_one_real )
& ( ord_less_real @ zero_zero_real @ ( set_sel_aux_complex @ F @ X3 @ Y ) ) ) ) ) ).
% set_sel_aux_reasonable_fin
thf(fact_855_set__sel__aux_H__reasonable__fin,axiom,
! [X3: set_nat,F: nat > nat > real,Y: nat] :
( ( finite_finite_nat @ X3 )
=> ( ( sel_reasonable_nat @ F )
=> ( ( ord_less_eq_real @ ( set_sel_aux_nat2 @ F @ X3 @ Y ) @ one_one_real )
& ( ord_less_real @ zero_zero_real @ ( set_sel_aux_nat2 @ F @ X3 @ Y ) ) ) ) ) ).
% set_sel_aux'_reasonable_fin
thf(fact_856_set__sel__aux_H__reasonable__fin,axiom,
! [X3: set_complex,F: complex > complex > real,Y: complex] :
( ( finite3207457112153483333omplex @ X3 )
=> ( ( sel_re4140149273989857872omplex @ F )
=> ( ( ord_less_eq_real @ ( set_sel_aux_complex2 @ F @ X3 @ Y ) @ one_one_real )
& ( ord_less_real @ zero_zero_real @ ( set_sel_aux_complex2 @ F @ X3 @ Y ) ) ) ) ) ).
% set_sel_aux'_reasonable_fin
thf(fact_857_List_Ofinite__set,axiom,
! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).
% List.finite_set
thf(fact_858_List_Ofinite__set,axiom,
! [Xs: list_complex] : ( finite3207457112153483333omplex @ ( set_complex2 @ Xs ) ) ).
% List.finite_set
thf(fact_859_finite__distinct__list,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ? [Xs3: list_nat] :
( ( ( set_nat2 @ Xs3 )
= A3 )
& ( distinct_nat @ Xs3 ) ) ) ).
% finite_distinct_list
thf(fact_860_finite__distinct__list,axiom,
! [A3: set_complex] :
( ( finite3207457112153483333omplex @ A3 )
=> ? [Xs3: list_complex] :
( ( ( set_complex2 @ Xs3 )
= A3 )
& ( distinct_complex @ Xs3 ) ) ) ).
% finite_distinct_list
thf(fact_861_finite__trans,axiom,
! [L: joinTree_nat,R2: joinTree_nat] :
( ( finite_finite_nat @ ( relations_nat @ ( join_nat @ L @ R2 ) ) )
=> ( ( finite_finite_nat @ ( relations_nat @ L ) )
& ( finite_finite_nat @ ( relations_nat @ R2 ) ) ) ) ).
% finite_trans
thf(fact_862_finite__trans,axiom,
! [L: joinTree_complex,R2: joinTree_complex] :
( ( finite3207457112153483333omplex @ ( relations_complex @ ( join_complex @ L @ R2 ) ) )
=> ( ( finite3207457112153483333omplex @ ( relations_complex @ L ) )
& ( finite3207457112153483333omplex @ ( relations_complex @ R2 ) ) ) ) ).
% finite_trans
thf(fact_863_finite__trans,axiom,
! [L: joinTree_a,R2: joinTree_a] :
( ( finite_finite_a @ ( relations_a @ ( join_a @ L @ R2 ) ) )
=> ( ( finite_finite_a @ ( relations_a @ L ) )
& ( finite_finite_a @ ( relations_a @ R2 ) ) ) ) ).
% finite_trans
thf(fact_864_finite__list,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ? [Xs3: list_nat] :
( ( set_nat2 @ Xs3 )
= A3 ) ) ).
% finite_list
thf(fact_865_finite__list,axiom,
! [A3: set_complex] :
( ( finite3207457112153483333omplex @ A3 )
=> ? [Xs3: list_complex] :
( ( set_complex2 @ Xs3 )
= A3 ) ) ).
% finite_list
thf(fact_866_set__sel_H__1__if__notfin2,axiom,
! [Y3: set_nat,F: nat > nat > real,X5: set_nat] :
( ~ ( finite_finite_nat @ Y3 )
=> ( ( set_sel_nat2 @ F @ X5 @ Y3 )
= one_one_real ) ) ).
% set_sel'_1_if_notfin2
thf(fact_867_set__sel_H__1__if__notfin2,axiom,
! [Y3: set_complex,F: complex > complex > real,X5: set_complex] :
( ~ ( finite3207457112153483333omplex @ Y3 )
=> ( ( set_sel_complex2 @ F @ X5 @ Y3 )
= one_one_real ) ) ).
% set_sel'_1_if_notfin2
thf(fact_868_set__sel_H__1__if__notfin1,axiom,
! [X5: set_nat,F: nat > nat > real,Y3: set_nat] :
( ~ ( finite_finite_nat @ X5 )
=> ( ( set_sel_nat2 @ F @ X5 @ Y3 )
= one_one_real ) ) ).
% set_sel'_1_if_notfin1
thf(fact_869_set__sel_H__1__if__notfin1,axiom,
! [X5: set_complex,F: complex > complex > real,Y3: set_complex] :
( ~ ( finite3207457112153483333omplex @ X5 )
=> ( ( set_sel_complex2 @ F @ X5 @ Y3 )
= one_one_real ) ) ).
% set_sel'_1_if_notfin1
thf(fact_870_set__sel__1__if__notfin2,axiom,
! [Y3: set_nat,F: nat > nat > real,X5: set_nat] :
( ~ ( finite_finite_nat @ Y3 )
=> ( ( set_sel_nat @ F @ X5 @ Y3 )
= one_one_real ) ) ).
% set_sel_1_if_notfin2
thf(fact_871_set__sel__1__if__notfin2,axiom,
! [Y3: set_complex,F: complex > complex > real,X5: set_complex] :
( ~ ( finite3207457112153483333omplex @ Y3 )
=> ( ( set_sel_complex @ F @ X5 @ Y3 )
= one_one_real ) ) ).
% set_sel_1_if_notfin2
thf(fact_872_set__sel__1__if__notfin1,axiom,
! [X5: set_nat,F: nat > nat > real,Y3: set_nat] :
( ~ ( finite_finite_nat @ X5 )
=> ( ( set_sel_nat @ F @ X5 @ Y3 )
= one_one_real ) ) ).
% set_sel_1_if_notfin1
thf(fact_873_set__sel__1__if__notfin1,axiom,
! [X5: set_complex,F: complex > complex > real,Y3: set_complex] :
( ~ ( finite3207457112153483333omplex @ X5 )
=> ( ( set_sel_complex @ F @ X5 @ Y3 )
= one_one_real ) ) ).
% set_sel_1_if_notfin1
thf(fact_874_set__sel__aux_H__1__if__notfin,axiom,
! [X5: set_nat,F: nat > nat > real,Y: nat] :
( ~ ( finite_finite_nat @ X5 )
=> ( ( set_sel_aux_nat2 @ F @ X5 @ Y )
= one_one_real ) ) ).
% set_sel_aux'_1_if_notfin
thf(fact_875_set__sel__aux_H__1__if__notfin,axiom,
! [X5: set_complex,F: complex > complex > real,Y: complex] :
( ~ ( finite3207457112153483333omplex @ X5 )
=> ( ( set_sel_aux_complex2 @ F @ X5 @ Y )
= one_one_real ) ) ).
% set_sel_aux'_1_if_notfin
thf(fact_876_set__sel__aux__1__if__notfin,axiom,
! [Y3: set_nat,F: nat > nat > real,X3: nat] :
( ~ ( finite_finite_nat @ Y3 )
=> ( ( set_sel_aux_nat @ F @ X3 @ Y3 )
= one_one_real ) ) ).
% set_sel_aux_1_if_notfin
thf(fact_877_set__sel__aux__1__if__notfin,axiom,
! [Y3: set_complex,F: complex > complex > real,X3: complex] :
( ~ ( finite3207457112153483333omplex @ Y3 )
=> ( ( set_sel_aux_complex @ F @ X3 @ Y3 )
= one_one_real ) ) ).
% set_sel_aux_1_if_notfin
thf(fact_878_set__sel__symm__if__finite,axiom,
! [X5: set_nat,Y3: set_nat,F: nat > nat > real] :
( ( finite_finite_nat @ X5 )
=> ( ( finite_finite_nat @ Y3 )
=> ( ( sel_symm_nat @ F )
=> ( ( set_sel_nat @ F @ X5 @ Y3 )
= ( set_sel_nat @ F @ Y3 @ X5 ) ) ) ) ) ).
% set_sel_symm_if_finite
thf(fact_879_set__sel__symm__if__finite,axiom,
! [X5: set_complex,Y3: set_complex,F: complex > complex > real] :
( ( finite3207457112153483333omplex @ X5 )
=> ( ( finite3207457112153483333omplex @ Y3 )
=> ( ( sel_symm_complex @ F )
=> ( ( set_sel_complex @ F @ X5 @ Y3 )
= ( set_sel_complex @ F @ Y3 @ X5 ) ) ) ) ) ).
% set_sel_symm_if_finite
thf(fact_880_set__sel_H__symm__if__finite,axiom,
! [X5: set_nat,Y3: set_nat,F: nat > nat > real] :
( ( finite_finite_nat @ X5 )
=> ( ( finite_finite_nat @ Y3 )
=> ( ( sel_symm_nat @ F )
=> ( ( set_sel_nat2 @ F @ X5 @ Y3 )
= ( set_sel_nat2 @ F @ Y3 @ X5 ) ) ) ) ) ).
% set_sel'_symm_if_finite
thf(fact_881_set__sel_H__symm__if__finite,axiom,
! [X5: set_complex,Y3: set_complex,F: complex > complex > real] :
( ( finite3207457112153483333omplex @ X5 )
=> ( ( finite3207457112153483333omplex @ Y3 )
=> ( ( sel_symm_complex @ F )
=> ( ( set_sel_complex2 @ F @ X5 @ Y3 )
= ( set_sel_complex2 @ F @ Y3 @ X5 ) ) ) ) ) ).
% set_sel'_symm_if_finite
thf(fact_882_set__sel_H__reasonable__fin,axiom,
! [Y: set_nat,F: nat > nat > real,X3: set_nat] :
( ( finite_finite_nat @ Y )
=> ( ( sel_reasonable_nat @ F )
=> ( ( ord_less_eq_real @ ( set_sel_nat2 @ F @ X3 @ Y ) @ one_one_real )
& ( ord_less_real @ zero_zero_real @ ( set_sel_nat2 @ F @ X3 @ Y ) ) ) ) ) ).
% set_sel'_reasonable_fin
thf(fact_883_set__sel_H__reasonable__fin,axiom,
! [Y: set_complex,F: complex > complex > real,X3: set_complex] :
( ( finite3207457112153483333omplex @ Y )
=> ( ( sel_re4140149273989857872omplex @ F )
=> ( ( ord_less_eq_real @ ( set_sel_complex2 @ F @ X3 @ Y ) @ one_one_real )
& ( ord_less_real @ zero_zero_real @ ( set_sel_complex2 @ F @ X3 @ Y ) ) ) ) ) ).
% set_sel'_reasonable_fin
thf(fact_884_set__sel__reasonable__fin,axiom,
! [X3: set_nat,F: nat > nat > real,Y: set_nat] :
( ( finite_finite_nat @ X3 )
=> ( ( sel_reasonable_nat @ F )
=> ( ( ord_less_eq_real @ ( set_sel_nat @ F @ X3 @ Y ) @ one_one_real )
& ( ord_less_real @ zero_zero_real @ ( set_sel_nat @ F @ X3 @ Y ) ) ) ) ) ).
% set_sel_reasonable_fin
thf(fact_885_set__sel__reasonable__fin,axiom,
! [X3: set_complex,F: complex > complex > real,Y: set_complex] :
( ( finite3207457112153483333omplex @ X3 )
=> ( ( sel_re4140149273989857872omplex @ F )
=> ( ( ord_less_eq_real @ ( set_sel_complex @ F @ X3 @ Y ) @ one_one_real )
& ( ord_less_real @ zero_zero_real @ ( set_sel_complex @ F @ X3 @ Y ) ) ) ) ) ).
% set_sel_reasonable_fin
thf(fact_886_finite__Collect__subsets,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B5: set_nat] : ( ord_less_eq_set_nat @ B5 @ A3 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_887_finite__Collect__subsets,axiom,
! [A3: set_complex] :
( ( finite3207457112153483333omplex @ A3 )
=> ( finite6551019134538273531omplex
@ ( collect_set_complex
@ ^ [B5: set_complex] : ( ord_le211207098394363844omplex @ B5 @ A3 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_888_finite__Collect__disjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( P @ X4 )
| ( Q @ X4 ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_889_finite__Collect__disjI,axiom,
! [P: complex > $o,Q: complex > $o] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X4: complex] :
( ( P @ X4 )
| ( Q @ X4 ) ) ) )
= ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
& ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_890_finite__Collect__conjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P ) )
| ( finite_finite_nat @ ( collect_nat @ Q ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( P @ X4 )
& ( Q @ X4 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_891_finite__Collect__conjI,axiom,
! [P: complex > $o,Q: complex > $o] :
( ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
| ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X4: complex] :
( ( P @ X4 )
& ( Q @ X4 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_892_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_1: nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_893_not__finite__existsD,axiom,
! [P: complex > $o] :
( ~ ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
=> ? [X_1: complex] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_894_pigeonhole__infinite__rel,axiom,
! [A3: set_nat,B3: set_nat,R4: nat > nat > $o] :
( ~ ( finite_finite_nat @ A3 )
=> ( ( finite_finite_nat @ B3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A3 )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ B3 )
& ( R4 @ X2 @ Xa2 ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B3 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A2: nat] :
( ( member_nat @ A2 @ A3 )
& ( R4 @ A2 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_895_pigeonhole__infinite__rel,axiom,
! [A3: set_nat,B3: set_complex,R4: nat > complex > $o] :
( ~ ( finite_finite_nat @ A3 )
=> ( ( finite3207457112153483333omplex @ B3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A3 )
=> ? [Xa2: complex] :
( ( member_complex @ Xa2 @ B3 )
& ( R4 @ X2 @ Xa2 ) ) )
=> ? [X2: complex] :
( ( member_complex @ X2 @ B3 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A2: nat] :
( ( member_nat @ A2 @ A3 )
& ( R4 @ A2 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_896_pigeonhole__infinite__rel,axiom,
! [A3: set_complex,B3: set_nat,R4: complex > nat > $o] :
( ~ ( finite3207457112153483333omplex @ A3 )
=> ( ( finite_finite_nat @ B3 )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ A3 )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ B3 )
& ( R4 @ X2 @ Xa2 ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B3 )
& ~ ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [A2: complex] :
( ( member_complex @ A2 @ A3 )
& ( R4 @ A2 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_897_pigeonhole__infinite__rel,axiom,
! [A3: set_complex,B3: set_complex,R4: complex > complex > $o] :
( ~ ( finite3207457112153483333omplex @ A3 )
=> ( ( finite3207457112153483333omplex @ B3 )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ A3 )
=> ? [Xa2: complex] :
( ( member_complex @ Xa2 @ B3 )
& ( R4 @ X2 @ Xa2 ) ) )
=> ? [X2: complex] :
( ( member_complex @ X2 @ B3 )
& ~ ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [A2: complex] :
( ( member_complex @ A2 @ A3 )
& ( R4 @ A2 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_898_prod_Ofinite__Collect__op,axiom,
! [I2: set_nat,X3: nat > nat,Y: nat > nat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I2 )
& ( ( X3 @ I3 )
!= one_one_nat ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I2 )
& ( ( Y @ I3 )
!= one_one_nat ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I2 )
& ( ( times_times_nat @ ( X3 @ I3 ) @ ( Y @ I3 ) )
!= one_one_nat ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_899_prod_Ofinite__Collect__op,axiom,
! [I2: set_complex,X3: complex > nat,Y: complex > nat] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I3: complex] :
( ( member_complex @ I3 @ I2 )
& ( ( X3 @ I3 )
!= one_one_nat ) ) ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I3: complex] :
( ( member_complex @ I3 @ I2 )
& ( ( Y @ I3 )
!= one_one_nat ) ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I3: complex] :
( ( member_complex @ I3 @ I2 )
& ( ( times_times_nat @ ( X3 @ I3 ) @ ( Y @ I3 ) )
!= one_one_nat ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_900_prod_Ofinite__Collect__op,axiom,
! [I2: set_nat,X3: nat > complex,Y: nat > complex] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I2 )
& ( ( X3 @ I3 )
!= one_one_complex ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I2 )
& ( ( Y @ I3 )
!= one_one_complex ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I2 )
& ( ( times_times_complex @ ( X3 @ I3 ) @ ( Y @ I3 ) )
!= one_one_complex ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_901_prod_Ofinite__Collect__op,axiom,
! [I2: set_complex,X3: complex > complex,Y: complex > complex] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I3: complex] :
( ( member_complex @ I3 @ I2 )
& ( ( X3 @ I3 )
!= one_one_complex ) ) ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I3: complex] :
( ( member_complex @ I3 @ I2 )
& ( ( Y @ I3 )
!= one_one_complex ) ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I3: complex] :
( ( member_complex @ I3 @ I2 )
& ( ( times_times_complex @ ( X3 @ I3 ) @ ( Y @ I3 ) )
!= one_one_complex ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_902_prod_Ofinite__Collect__op,axiom,
! [I2: set_nat,X3: nat > real,Y: nat > real] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I2 )
& ( ( X3 @ I3 )
!= one_one_real ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I2 )
& ( ( Y @ I3 )
!= one_one_real ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I2 )
& ( ( times_times_real @ ( X3 @ I3 ) @ ( Y @ I3 ) )
!= one_one_real ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_903_prod_Ofinite__Collect__op,axiom,
! [I2: set_complex,X3: complex > real,Y: complex > real] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I3: complex] :
( ( member_complex @ I3 @ I2 )
& ( ( X3 @ I3 )
!= one_one_real ) ) ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I3: complex] :
( ( member_complex @ I3 @ I2 )
& ( ( Y @ I3 )
!= one_one_real ) ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I3: complex] :
( ( member_complex @ I3 @ I2 )
& ( ( times_times_real @ ( X3 @ I3 ) @ ( Y @ I3 ) )
!= one_one_real ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_904_sum_Ofinite__Collect__op,axiom,
! [I2: set_nat,X3: nat > real,Y: nat > real] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I2 )
& ( ( X3 @ I3 )
!= zero_zero_real ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I2 )
& ( ( Y @ I3 )
!= zero_zero_real ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I2 )
& ( ( plus_plus_real @ ( X3 @ I3 ) @ ( Y @ I3 ) )
!= zero_zero_real ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_905_sum_Ofinite__Collect__op,axiom,
! [I2: set_complex,X3: complex > real,Y: complex > real] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I3: complex] :
( ( member_complex @ I3 @ I2 )
& ( ( X3 @ I3 )
!= zero_zero_real ) ) ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I3: complex] :
( ( member_complex @ I3 @ I2 )
& ( ( Y @ I3 )
!= zero_zero_real ) ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I3: complex] :
( ( member_complex @ I3 @ I2 )
& ( ( plus_plus_real @ ( X3 @ I3 ) @ ( Y @ I3 ) )
!= zero_zero_real ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_906_sum_Ofinite__Collect__op,axiom,
! [I2: set_nat,X3: nat > nat,Y: nat > nat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I2 )
& ( ( X3 @ I3 )
!= zero_zero_nat ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I2 )
& ( ( Y @ I3 )
!= zero_zero_nat ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I2 )
& ( ( plus_plus_nat @ ( X3 @ I3 ) @ ( Y @ I3 ) )
!= zero_zero_nat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_907_sum_Ofinite__Collect__op,axiom,
! [I2: set_complex,X3: complex > nat,Y: complex > nat] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I3: complex] :
( ( member_complex @ I3 @ I2 )
& ( ( X3 @ I3 )
!= zero_zero_nat ) ) ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I3: complex] :
( ( member_complex @ I3 @ I2 )
& ( ( Y @ I3 )
!= zero_zero_nat ) ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I3: complex] :
( ( member_complex @ I3 @ I2 )
& ( ( plus_plus_nat @ ( X3 @ I3 ) @ ( Y @ I3 ) )
!= zero_zero_nat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_908_sum_Ofinite__Collect__op,axiom,
! [I2: set_nat,X3: nat > complex,Y: nat > complex] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I2 )
& ( ( X3 @ I3 )
!= zero_zero_complex ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I2 )
& ( ( Y @ I3 )
!= zero_zero_complex ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I2 )
& ( ( plus_plus_complex @ ( X3 @ I3 ) @ ( Y @ I3 ) )
!= zero_zero_complex ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_909_sum_Ofinite__Collect__op,axiom,
! [I2: set_complex,X3: complex > complex,Y: complex > complex] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I3: complex] :
( ( member_complex @ I3 @ I2 )
& ( ( X3 @ I3 )
!= zero_zero_complex ) ) ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I3: complex] :
( ( member_complex @ I3 @ I2 )
& ( ( Y @ I3 )
!= zero_zero_complex ) ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I3: complex] :
( ( member_complex @ I3 @ I2 )
& ( ( plus_plus_complex @ ( X3 @ I3 ) @ ( Y @ I3 ) )
!= zero_zero_complex ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_910_filter__preserves__multiset,axiom,
! [M2: nat > nat,P: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] : ( ord_less_nat @ zero_zero_nat @ ( M2 @ X4 ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X4 ) @ ( M2 @ X4 ) @ zero_zero_nat ) ) ) ) ) ).
% filter_preserves_multiset
thf(fact_911_filter__preserves__multiset,axiom,
! [M2: complex > nat,P: complex > $o] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X4: complex] : ( ord_less_nat @ zero_zero_nat @ ( M2 @ X4 ) ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X4: complex] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X4 ) @ ( M2 @ X4 ) @ zero_zero_nat ) ) ) ) ) ).
% filter_preserves_multiset
thf(fact_912_field__lbound__gt__zero,axiom,
! [D1: real,D2: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D2 )
=> ? [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
& ( ord_less_real @ E2 @ D1 )
& ( ord_less_real @ E2 @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_913_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_nat @ N2 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_914_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_915_add__mset__in__multiset,axiom,
! [M2: nat > nat,A: nat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] : ( ord_less_nat @ zero_zero_nat @ ( M2 @ X4 ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X4 = A ) @ ( suc @ ( M2 @ X4 ) ) @ ( M2 @ X4 ) ) ) ) ) ) ).
% add_mset_in_multiset
thf(fact_916_add__mset__in__multiset,axiom,
! [M2: complex > nat,A: complex] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X4: complex] : ( ord_less_nat @ zero_zero_nat @ ( M2 @ X4 ) ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X4: complex] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X4 = A ) @ ( suc @ ( M2 @ X4 ) ) @ ( M2 @ X4 ) ) ) ) ) ) ).
% add_mset_in_multiset
thf(fact_917_ldeep__s__last1__if__distinct,axiom,
! [Xs: list_a,Sel: a > a > real] :
( ( distinct_a @ Xs )
=> ( ( ldeep_s_a @ Sel @ Xs @ ( last_a @ Xs ) )
= one_one_real ) ) ).
% ldeep_s_last1_if_distinct
thf(fact_918_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K2: nat] :
( ( P @ K2 )
& ( ord_less_nat @ K2 @ I ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_919_joinTree_Osize__gen_I2_J,axiom,
! [X3: a > nat,X21: joinTree_a,X22: joinTree_a] :
( ( size_joinTree_a @ X3 @ ( join_a @ X21 @ X22 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_joinTree_a @ X3 @ X21 ) @ ( size_joinTree_a @ X3 @ X22 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% joinTree.size_gen(2)
thf(fact_920_finite__less__ub,axiom,
! [F: nat > nat,U: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( F @ N3 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_921_joinTree_Osize__gen_I1_J,axiom,
! [X3: a > nat,X1: a] :
( ( size_joinTree_a @ X3 @ ( relation_a @ X1 ) )
= ( plus_plus_nat @ ( X3 @ X1 ) @ ( suc @ zero_zero_nat ) ) ) ).
% joinTree.size_gen(1)
thf(fact_922_joinTree_Osize_I4_J,axiom,
! [X21: joinTree_a,X22: joinTree_a] :
( ( size_size_joinTree_a @ ( join_a @ X21 @ X22 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_size_joinTree_a @ X21 ) @ ( size_size_joinTree_a @ X22 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% joinTree.size(4)
thf(fact_923_joinTree_Osize_I3_J,axiom,
! [X1: a] :
( ( size_size_joinTree_a @ ( relation_a @ X1 ) )
= ( suc @ zero_zero_nat ) ) ).
% joinTree.size(3)
thf(fact_924_power__decreasing__iff,axiom,
! [B: real,M: nat,N: nat] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_real @ B @ one_one_real )
=> ( ( ord_less_eq_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_925_power__decreasing__iff,axiom,
! [B: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_926_ln__powr__bound,axiom,
! [X3: real,A: real] :
( ( ord_less_eq_real @ one_one_real @ X3 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( ln_ln_real @ X3 ) @ ( divide_divide_real @ ( powr_real @ X3 @ A ) @ A ) ) ) ) ).
% ln_powr_bound
thf(fact_927_power__one__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_928_power__one__right,axiom,
! [A: real] :
( ( power_power_real @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_929_power__one__right,axiom,
! [A: complex] :
( ( power_power_complex @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_930_power__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% power_one
thf(fact_931_power__one,axiom,
! [N: nat] :
( ( power_power_real @ one_one_real @ N )
= one_one_real ) ).
% power_one
thf(fact_932_power__one,axiom,
! [N: nat] :
( ( power_power_complex @ one_one_complex @ N )
= one_one_complex ) ).
% power_one
thf(fact_933_nat__power__eq__Suc__0__iff,axiom,
! [X3: nat,M: nat] :
( ( ( power_power_nat @ X3 @ M )
= ( suc @ zero_zero_nat ) )
= ( ( M = zero_zero_nat )
| ( X3
= ( suc @ zero_zero_nat ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_934_power__Suc__0,axiom,
! [N: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_935_nat__zero__less__power__iff,axiom,
! [X3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X3 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X3 )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_936_powr__0,axiom,
! [Z: real] :
( ( powr_real @ zero_zero_real @ Z )
= zero_zero_real ) ).
% powr_0
thf(fact_937_powr__eq__0__iff,axiom,
! [W: real,Z: real] :
( ( ( powr_real @ W @ Z )
= zero_zero_real )
= ( W = zero_zero_real ) ) ).
% powr_eq_0_iff
thf(fact_938_power__inject__exp,axiom,
! [A: real,M: nat,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ( power_power_real @ A @ M )
= ( power_power_real @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_939_power__inject__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ( power_power_nat @ A @ M )
= ( power_power_nat @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_940_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
= zero_zero_nat ) ).
% power_0_Suc
thf(fact_941_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
= zero_zero_real ) ).
% power_0_Suc
thf(fact_942_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_complex @ zero_zero_complex @ ( suc @ N ) )
= zero_zero_complex ) ).
% power_0_Suc
thf(fact_943_power__Suc0__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_944_power__Suc0__right,axiom,
! [A: real] :
( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_945_power__Suc0__right,axiom,
! [A: complex] :
( ( power_power_complex @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_946_powr__zero__eq__one,axiom,
! [X3: real] :
( ( ( X3 = zero_zero_real )
=> ( ( powr_real @ X3 @ zero_zero_real )
= zero_zero_real ) )
& ( ( X3 != zero_zero_real )
=> ( ( powr_real @ X3 @ zero_zero_real )
= one_one_real ) ) ) ).
% powr_zero_eq_one
thf(fact_947_powr__gt__zero,axiom,
! [X3: real,A: real] :
( ( ord_less_real @ zero_zero_real @ ( powr_real @ X3 @ A ) )
= ( X3 != zero_zero_real ) ) ).
% powr_gt_zero
thf(fact_948_powr__nonneg__iff,axiom,
! [A: real,X3: real] :
( ( ord_less_eq_real @ ( powr_real @ A @ X3 ) @ zero_zero_real )
= ( A = zero_zero_real ) ) ).
% powr_nonneg_iff
thf(fact_949_power__strict__increasing__iff,axiom,
! [B: real,X3: nat,Y: nat] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ ( power_power_real @ B @ X3 ) @ ( power_power_real @ B @ Y ) )
= ( ord_less_nat @ X3 @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_950_power__strict__increasing__iff,axiom,
! [B: nat,X3: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ X3 ) @ ( power_power_nat @ B @ Y ) )
= ( ord_less_nat @ X3 @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_951_power__eq__0__iff,axiom,
! [A: nat,N: nat] :
( ( ( power_power_nat @ A @ N )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_952_power__eq__0__iff,axiom,
! [A: real,N: nat] :
( ( ( power_power_real @ A @ N )
= zero_zero_real )
= ( ( A = zero_zero_real )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_953_power__eq__0__iff,axiom,
! [A: complex,N: nat] :
( ( ( power_power_complex @ A @ N )
= zero_zero_complex )
= ( ( A = zero_zero_complex )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_954_powr__eq__one__iff,axiom,
! [A: real,X3: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ( powr_real @ A @ X3 )
= one_one_real )
= ( X3 = zero_zero_real ) ) ) ).
% powr_eq_one_iff
thf(fact_955_powr__one__gt__zero__iff,axiom,
! [X3: real] :
( ( ( powr_real @ X3 @ one_one_real )
= X3 )
= ( ord_less_eq_real @ zero_zero_real @ X3 ) ) ).
% powr_one_gt_zero_iff
thf(fact_956_powr__one,axiom,
! [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( powr_real @ X3 @ one_one_real )
= X3 ) ) ).
% powr_one
thf(fact_957_power__strict__decreasing__iff,axiom,
! [B: real,M: nat,N: nat] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_real @ B @ one_one_real )
=> ( ( ord_less_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_958_power__strict__decreasing__iff,axiom,
! [B: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_959_power__increasing__iff,axiom,
! [B: real,X3: nat,Y: nat] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_eq_real @ ( power_power_real @ B @ X3 ) @ ( power_power_real @ B @ Y ) )
= ( ord_less_eq_nat @ X3 @ Y ) ) ) ).
% power_increasing_iff
thf(fact_960_power__increasing__iff,axiom,
! [B: nat,X3: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X3 ) @ ( power_power_nat @ B @ Y ) )
= ( ord_less_eq_nat @ X3 @ Y ) ) ) ).
% power_increasing_iff
thf(fact_961_power__mono__iff,axiom,
! [A: real,B: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
= ( ord_less_eq_real @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_962_power__mono__iff,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_963_power__Suc,axiom,
! [A: nat,N: nat] :
( ( power_power_nat @ A @ ( suc @ N ) )
= ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% power_Suc
thf(fact_964_power__Suc,axiom,
! [A: complex,N: nat] :
( ( power_power_complex @ A @ ( suc @ N ) )
= ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).
% power_Suc
thf(fact_965_power__Suc,axiom,
! [A: real,N: nat] :
( ( power_power_real @ A @ ( suc @ N ) )
= ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).
% power_Suc
thf(fact_966_power__Suc2,axiom,
! [A: nat,N: nat] :
( ( power_power_nat @ A @ ( suc @ N ) )
= ( times_times_nat @ ( power_power_nat @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_967_power__Suc2,axiom,
! [A: complex,N: nat] :
( ( power_power_complex @ A @ ( suc @ N ) )
= ( times_times_complex @ ( power_power_complex @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_968_power__Suc2,axiom,
! [A: real,N: nat] :
( ( power_power_real @ A @ ( suc @ N ) )
= ( times_times_real @ ( power_power_real @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_969_power__mult,axiom,
! [A: nat,M: nat,N: nat] :
( ( power_power_nat @ A @ ( times_times_nat @ M @ N ) )
= ( power_power_nat @ ( power_power_nat @ A @ M ) @ N ) ) ).
% power_mult
thf(fact_970_power__mult,axiom,
! [A: real,M: nat,N: nat] :
( ( power_power_real @ A @ ( times_times_nat @ M @ N ) )
= ( power_power_real @ ( power_power_real @ A @ M ) @ N ) ) ).
% power_mult
thf(fact_971_power__mult,axiom,
! [A: complex,M: nat,N: nat] :
( ( power_power_complex @ A @ ( times_times_nat @ M @ N ) )
= ( power_power_complex @ ( power_power_complex @ A @ M ) @ N ) ) ).
% power_mult
thf(fact_972_power__divide,axiom,
! [A: complex,B: complex,N: nat] :
( ( power_power_complex @ ( divide1717551699836669952omplex @ A @ B ) @ N )
= ( divide1717551699836669952omplex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B @ N ) ) ) ).
% power_divide
thf(fact_973_power__divide,axiom,
! [A: real,B: real,N: nat] :
( ( power_power_real @ ( divide_divide_real @ A @ B ) @ N )
= ( divide_divide_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).
% power_divide
thf(fact_974_power__commuting__commutes,axiom,
! [X3: nat,Y: nat,N: nat] :
( ( ( times_times_nat @ X3 @ Y )
= ( times_times_nat @ Y @ X3 ) )
=> ( ( times_times_nat @ ( power_power_nat @ X3 @ N ) @ Y )
= ( times_times_nat @ Y @ ( power_power_nat @ X3 @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_975_power__commuting__commutes,axiom,
! [X3: complex,Y: complex,N: nat] :
( ( ( times_times_complex @ X3 @ Y )
= ( times_times_complex @ Y @ X3 ) )
=> ( ( times_times_complex @ ( power_power_complex @ X3 @ N ) @ Y )
= ( times_times_complex @ Y @ ( power_power_complex @ X3 @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_976_power__commuting__commutes,axiom,
! [X3: real,Y: real,N: nat] :
( ( ( times_times_real @ X3 @ Y )
= ( times_times_real @ Y @ X3 ) )
=> ( ( times_times_real @ ( power_power_real @ X3 @ N ) @ Y )
= ( times_times_real @ Y @ ( power_power_real @ X3 @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_977_power__mult__distrib,axiom,
! [A: nat,B: nat,N: nat] :
( ( power_power_nat @ ( times_times_nat @ A @ B ) @ N )
= ( times_times_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_978_power__mult__distrib,axiom,
! [A: complex,B: complex,N: nat] :
( ( power_power_complex @ ( times_times_complex @ A @ B ) @ N )
= ( times_times_complex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_979_power__mult__distrib,axiom,
! [A: real,B: real,N: nat] :
( ( power_power_real @ ( times_times_real @ A @ B ) @ N )
= ( times_times_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_980_power__commutes,axiom,
! [A: nat,N: nat] :
( ( times_times_nat @ ( power_power_nat @ A @ N ) @ A )
= ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% power_commutes
thf(fact_981_power__commutes,axiom,
! [A: complex,N: nat] :
( ( times_times_complex @ ( power_power_complex @ A @ N ) @ A )
= ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).
% power_commutes
thf(fact_982_power__commutes,axiom,
! [A: real,N: nat] :
( ( times_times_real @ ( power_power_real @ A @ N ) @ A )
= ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).
% power_commutes
thf(fact_983_power__not__zero,axiom,
! [A: nat,N: nat] :
( ( A != zero_zero_nat )
=> ( ( power_power_nat @ A @ N )
!= zero_zero_nat ) ) ).
% power_not_zero
thf(fact_984_power__not__zero,axiom,
! [A: real,N: nat] :
( ( A != zero_zero_real )
=> ( ( power_power_real @ A @ N )
!= zero_zero_real ) ) ).
% power_not_zero
thf(fact_985_power__not__zero,axiom,
! [A: complex,N: nat] :
( ( A != zero_zero_complex )
=> ( ( power_power_complex @ A @ N )
!= zero_zero_complex ) ) ).
% power_not_zero
thf(fact_986_powr__powr,axiom,
! [X3: real,A: real,B: real] :
( ( powr_real @ ( powr_real @ X3 @ A ) @ B )
= ( powr_real @ X3 @ ( times_times_real @ A @ B ) ) ) ).
% powr_powr
thf(fact_987_power__0,axiom,
! [A: nat] :
( ( power_power_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_988_power__0,axiom,
! [A: real] :
( ( power_power_real @ A @ zero_zero_nat )
= one_one_real ) ).
% power_0
thf(fact_989_power__0,axiom,
! [A: complex] :
( ( power_power_complex @ A @ zero_zero_nat )
= one_one_complex ) ).
% power_0
thf(fact_990_power__one__over,axiom,
! [A: complex,N: nat] :
( ( power_power_complex @ ( divide1717551699836669952omplex @ one_one_complex @ A ) @ N )
= ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ A @ N ) ) ) ).
% power_one_over
thf(fact_991_power__one__over,axiom,
! [A: real,N: nat] :
( ( power_power_real @ ( divide_divide_real @ one_one_real @ A ) @ N )
= ( divide_divide_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% power_one_over
thf(fact_992_left__right__inverse__power,axiom,
! [X3: nat,Y: nat,N: nat] :
( ( ( times_times_nat @ X3 @ Y )
= one_one_nat )
=> ( ( times_times_nat @ ( power_power_nat @ X3 @ N ) @ ( power_power_nat @ Y @ N ) )
= one_one_nat ) ) ).
% left_right_inverse_power
thf(fact_993_left__right__inverse__power,axiom,
! [X3: complex,Y: complex,N: nat] :
( ( ( times_times_complex @ X3 @ Y )
= one_one_complex )
=> ( ( times_times_complex @ ( power_power_complex @ X3 @ N ) @ ( power_power_complex @ Y @ N ) )
= one_one_complex ) ) ).
% left_right_inverse_power
thf(fact_994_left__right__inverse__power,axiom,
! [X3: real,Y: real,N: nat] :
( ( ( times_times_real @ X3 @ Y )
= one_one_real )
=> ( ( times_times_real @ ( power_power_real @ X3 @ N ) @ ( power_power_real @ Y @ N ) )
= one_one_real ) ) ).
% left_right_inverse_power
thf(fact_995_nat__power__less__imp__less,axiom,
! [I: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_996_powr__less__mono2__neg,axiom,
! [A: real,X3: real,Y: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ X3 @ Y )
=> ( ord_less_real @ ( powr_real @ Y @ A ) @ ( powr_real @ X3 @ A ) ) ) ) ) ).
% powr_less_mono2_neg
thf(fact_997_powr__non__neg,axiom,
! [A: real,X3: real] :
~ ( ord_less_real @ ( powr_real @ A @ X3 ) @ zero_zero_real ) ).
% powr_non_neg
thf(fact_998_zero__less__power,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% zero_less_power
thf(fact_999_zero__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_less_power
thf(fact_1000_powr__ge__pzero,axiom,
! [X3: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( powr_real @ X3 @ Y ) ) ).
% powr_ge_pzero
thf(fact_1001_powr__mono2,axiom,
! [A: real,X3: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ X3 @ Y )
=> ( ord_less_eq_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).
% powr_mono2
thf(fact_1002_power__mono,axiom,
! [A: real,B: real,N: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% power_mono
thf(fact_1003_power__mono,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% power_mono
thf(fact_1004_zero__le__power,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% zero_le_power
thf(fact_1005_zero__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_le_power
thf(fact_1006_one__le__power,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ one_one_real @ A )
=> ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% one_le_power
thf(fact_1007_one__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).
% one_le_power
thf(fact_1008_power__add,axiom,
! [A: nat,M: nat,N: nat] :
( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% power_add
thf(fact_1009_power__add,axiom,
! [A: complex,M: nat,N: nat] :
( ( power_power_complex @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ).
% power_add
thf(fact_1010_power__add,axiom,
! [A: real,M: nat,N: nat] :
( ( power_power_real @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).
% power_add
thf(fact_1011_powr__mono2_H,axiom,
! [A: real,X3: real,Y: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ X3 @ Y )
=> ( ord_less_eq_real @ ( powr_real @ Y @ A ) @ ( powr_real @ X3 @ A ) ) ) ) ) ).
% powr_mono2'
thf(fact_1012_powr__less__mono2,axiom,
! [A: real,X3: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_real @ X3 @ Y )
=> ( ord_less_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).
% powr_less_mono2
thf(fact_1013_gr__one__powr,axiom,
! [X3: real,Y: real] :
( ( ord_less_real @ one_one_real @ X3 )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_real @ one_one_real @ ( powr_real @ X3 @ Y ) ) ) ) ).
% gr_one_powr
thf(fact_1014_powr__inj,axiom,
! [A: real,X3: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( ( powr_real @ A @ X3 )
= ( powr_real @ A @ Y ) )
= ( X3 = Y ) ) ) ) ).
% powr_inj
thf(fact_1015_ge__one__powr__ge__zero,axiom,
! [X3: real,A: real] :
( ( ord_less_eq_real @ one_one_real @ X3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ one_one_real @ ( powr_real @ X3 @ A ) ) ) ) ).
% ge_one_powr_ge_zero
thf(fact_1016_powr__mono__both,axiom,
! [A: real,B: real,X3: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ one_one_real @ X3 )
=> ( ( ord_less_eq_real @ X3 @ Y )
=> ( ord_less_eq_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ Y @ B ) ) ) ) ) ) ).
% powr_mono_both
thf(fact_1017_powr__le1,axiom,
! [A: real,X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ X3 @ one_one_real )
=> ( ord_less_eq_real @ ( powr_real @ X3 @ A ) @ one_one_real ) ) ) ) ).
% powr_le1
thf(fact_1018_powr__mult,axiom,
! [X3: real,Y: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( powr_real @ ( times_times_real @ X3 @ Y ) @ A )
= ( times_times_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).
% powr_mult
thf(fact_1019_powr__divide,axiom,
! [X3: real,Y: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( powr_real @ ( divide_divide_real @ X3 @ Y ) @ A )
= ( divide_divide_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).
% powr_divide
thf(fact_1020_ln__powr,axiom,
! [X3: real,Y: real] :
( ( X3 != zero_zero_real )
=> ( ( ln_ln_real @ ( powr_real @ X3 @ Y ) )
= ( times_times_real @ Y @ ( ln_ln_real @ X3 ) ) ) ) ).
% ln_powr
thf(fact_1021_power__less__imp__less__base,axiom,
! [A: real,N: nat,B: real] :
( ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_1022_power__less__imp__less__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_1023_power__le__one,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ one_one_real ) ) ) ).
% power_le_one
thf(fact_1024_power__le__one,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ one_one_nat ) ) ) ).
% power_le_one
thf(fact_1025_power__le__imp__le__base,axiom,
! [A: real,N: nat,B: real] :
( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ ( power_power_real @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_1026_power__le__imp__le__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ ( power_power_nat @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_1027_power__inject__base,axiom,
! [A: real,N: nat,B: real] :
( ( ( power_power_real @ A @ ( suc @ N ) )
= ( power_power_real @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_1028_power__inject__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ( power_power_nat @ A @ ( suc @ N ) )
= ( power_power_nat @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_1029_power__gt1__lemma,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_1030_power__gt1__lemma,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_1031_power__less__power__Suc,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ ( power_power_real @ A @ N ) @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_1032_power__less__power__Suc,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_1033_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ) ).
% power_0_left
thf(fact_1034_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N )
= one_one_real ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N )
= zero_zero_real ) ) ) ).
% power_0_left
thf(fact_1035_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_complex @ zero_zero_complex @ N )
= one_one_complex ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_complex @ zero_zero_complex @ N )
= zero_zero_complex ) ) ) ).
% power_0_left
thf(fact_1036_power__gt1,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ one_one_real @ ( power_power_real @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_1037_power__gt1,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_1038_power__increasing,axiom,
! [N: nat,N4: nat,A: real] :
( ( ord_less_eq_nat @ N @ N4 )
=> ( ( ord_less_eq_real @ one_one_real @ A )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N4 ) ) ) ) ).
% power_increasing
thf(fact_1039_power__increasing,axiom,
! [N: nat,N4: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N4 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N4 ) ) ) ) ).
% power_increasing
thf(fact_1040_power__strict__increasing,axiom,
! [N: nat,N4: nat,A: real] :
( ( ord_less_nat @ N @ N4 )
=> ( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N4 ) ) ) ) ).
% power_strict_increasing
thf(fact_1041_power__strict__increasing,axiom,
! [N: nat,N4: nat,A: nat] :
( ( ord_less_nat @ N @ N4 )
=> ( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N4 ) ) ) ) ).
% power_strict_increasing
thf(fact_1042_power__less__imp__less__exp,axiom,
! [A: real,M: nat,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_1043_power__less__imp__less__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_1044_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ).
% zero_power
thf(fact_1045_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_real @ zero_zero_real @ N )
= zero_zero_real ) ) ).
% zero_power
thf(fact_1046_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_complex @ zero_zero_complex @ N )
= zero_zero_complex ) ) ).
% zero_power
thf(fact_1047_powr__add,axiom,
! [X3: real,A: real,B: real] :
( ( powr_real @ X3 @ ( plus_plus_real @ A @ B ) )
= ( times_times_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ X3 @ B ) ) ) ).
% powr_add
thf(fact_1048_power__gt__expt,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).
% power_gt_expt
thf(fact_1049_nat__one__le__power,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N ) ) ) ).
% nat_one_le_power
thf(fact_1050_realpow__pos__nth2,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ? [R: real] :
( ( ord_less_real @ zero_zero_real @ R )
& ( ( power_power_real @ R @ ( suc @ N ) )
= A ) ) ) ).
% realpow_pos_nth2
thf(fact_1051_real__arch__pow__inv,axiom,
! [Y: real,X3: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ X3 @ one_one_real )
=> ? [N3: nat] : ( ord_less_real @ ( power_power_real @ X3 @ N3 ) @ Y ) ) ) ).
% real_arch_pow_inv
thf(fact_1052_finite__roots__unity,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( finite_finite_real
@ ( collect_real
@ ^ [Z4: real] :
( ( power_power_real @ Z4 @ N )
= one_one_real ) ) ) ) ).
% finite_roots_unity
thf(fact_1053_finite__roots__unity,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [Z4: complex] :
( ( power_power_complex @ Z4 @ N )
= one_one_complex ) ) ) ) ).
% finite_roots_unity
thf(fact_1054_power__Suc__less,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) @ ( power_power_real @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_1055_power__Suc__less,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_1056_power__Suc__le__self,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_1057_power__Suc__le__self,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_1058_power__Suc__less__one,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ ( power_power_real @ A @ ( suc @ N ) ) @ one_one_real ) ) ) ).
% power_Suc_less_one
thf(fact_1059_power__Suc__less__one,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ one_one_nat ) ) ) ).
% power_Suc_less_one
thf(fact_1060_power__decreasing,axiom,
! [N: nat,N4: nat,A: real] :
( ( ord_less_eq_nat @ N @ N4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N4 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_1061_power__decreasing,axiom,
! [N: nat,N4: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N4 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_1062_power__strict__decreasing,axiom,
! [N: nat,N4: nat,A: real] :
( ( ord_less_nat @ N @ N4 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ ( power_power_real @ A @ N4 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_1063_power__strict__decreasing,axiom,
! [N: nat,N4: nat,A: nat] :
( ( ord_less_nat @ N @ N4 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ N4 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_1064_power__le__imp__le__exp,axiom,
! [A: real,M: nat,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_eq_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_1065_power__le__imp__le__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_1066_power__eq__iff__eq__base,axiom,
! [N: nat,A: real,B: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ( power_power_real @ A @ N )
= ( power_power_real @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_1067_power__eq__iff__eq__base,axiom,
! [N: nat,A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_1068_power__eq__imp__eq__base,axiom,
! [A: real,N: nat,B: real] :
( ( ( power_power_real @ A @ N )
= ( power_power_real @ B @ N ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_1069_power__eq__imp__eq__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_1070_self__le__power,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ one_one_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_1071_self__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_1072_one__less__power,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_1073_one__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_1074_realpow__pos__nth__unique,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [X2: real] :
( ( ord_less_real @ zero_zero_real @ X2 )
& ( ( power_power_real @ X2 @ N )
= A )
& ! [Y4: real] :
( ( ( ord_less_real @ zero_zero_real @ Y4 )
& ( ( power_power_real @ Y4 @ N )
= A ) )
=> ( Y4 = X2 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_1075_realpow__pos__nth,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [R: real] :
( ( ord_less_real @ zero_zero_real @ R )
& ( ( power_power_real @ R @ N )
= A ) ) ) ) ).
% realpow_pos_nth
thf(fact_1076_powr__mult__base,axiom,
! [X3: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( ( times_times_real @ X3 @ ( powr_real @ X3 @ Y ) )
= ( powr_real @ X3 @ ( plus_plus_real @ one_one_real @ Y ) ) ) ) ).
% powr_mult_base
thf(fact_1077_power__strict__mono,axiom,
! [A: real,B: real,N: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_1078_power__strict__mono,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_1079_ln__powr__bound2,axiom,
! [X3: real,A: real] :
( ( ord_less_real @ one_one_real @ X3 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( powr_real @ ( ln_ln_real @ X3 ) @ A ) @ ( times_times_real @ ( powr_real @ A @ A ) @ X3 ) ) ) ) ).
% ln_powr_bound2
thf(fact_1080_finite__nth__roots,axiom,
! [N: nat,C: complex] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [Z4: complex] :
( ( power_power_complex @ Z4 @ N )
= C ) ) ) ) ).
% finite_nth_roots
thf(fact_1081_sum__zero__power_H,axiom,
! [A3: set_nat,C: nat > complex,D: nat > complex] :
( ( ( ( finite_finite_nat @ A3 )
& ( member_nat @ zero_zero_nat @ A3 ) )
=> ( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ zero_zero_complex @ I3 ) ) @ ( D @ I3 ) )
@ A3 )
= ( divide1717551699836669952omplex @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
& ( ~ ( ( finite_finite_nat @ A3 )
& ( member_nat @ zero_zero_nat @ A3 ) )
=> ( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ zero_zero_complex @ I3 ) ) @ ( D @ I3 ) )
@ A3 )
= zero_zero_complex ) ) ) ).
% sum_zero_power'
thf(fact_1082_sum__zero__power_H,axiom,
! [A3: set_nat,C: nat > real,D: nat > real] :
( ( ( ( finite_finite_nat @ A3 )
& ( member_nat @ zero_zero_nat @ A3 ) )
=> ( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ zero_zero_real @ I3 ) ) @ ( D @ I3 ) )
@ A3 )
= ( divide_divide_real @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
& ( ~ ( ( finite_finite_nat @ A3 )
& ( member_nat @ zero_zero_nat @ A3 ) )
=> ( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ zero_zero_real @ I3 ) ) @ ( D @ I3 ) )
@ A3 )
= zero_zero_real ) ) ) ).
% sum_zero_power'
thf(fact_1083_sum_Oneutral__const,axiom,
! [A3: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [Uu3: complex] : zero_zero_complex
@ A3 )
= zero_zero_complex ) ).
% sum.neutral_const
thf(fact_1084_sum__eq__0__iff,axiom,
! [F4: set_nat,F: nat > nat] :
( ( finite_finite_nat @ F4 )
=> ( ( ( groups3542108847815614940at_nat @ F @ F4 )
= zero_zero_nat )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ F4 )
=> ( ( F @ X4 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_1085_sum__eq__0__iff,axiom,
! [F4: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ F4 )
=> ( ( ( groups5693394587270226106ex_nat @ F @ F4 )
= zero_zero_nat )
= ( ! [X4: complex] :
( ( member_complex @ X4 @ F4 )
=> ( ( F @ X4 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_1086_sum_Oinfinite,axiom,
! [A3: set_nat,G: nat > real] :
( ~ ( finite_finite_nat @ A3 )
=> ( ( groups6591440286371151544t_real @ G @ A3 )
= zero_zero_real ) ) ).
% sum.infinite
thf(fact_1087_sum_Oinfinite,axiom,
! [A3: set_complex,G: complex > real] :
( ~ ( finite3207457112153483333omplex @ A3 )
=> ( ( groups5808333547571424918x_real @ G @ A3 )
= zero_zero_real ) ) ).
% sum.infinite
thf(fact_1088_sum_Oinfinite,axiom,
! [A3: set_nat,G: nat > nat] :
( ~ ( finite_finite_nat @ A3 )
=> ( ( groups3542108847815614940at_nat @ G @ A3 )
= zero_zero_nat ) ) ).
% sum.infinite
thf(fact_1089_sum_Oinfinite,axiom,
! [A3: set_complex,G: complex > nat] :
( ~ ( finite3207457112153483333omplex @ A3 )
=> ( ( groups5693394587270226106ex_nat @ G @ A3 )
= zero_zero_nat ) ) ).
% sum.infinite
thf(fact_1090_sum_Oinfinite,axiom,
! [A3: set_nat,G: nat > complex] :
( ~ ( finite_finite_nat @ A3 )
=> ( ( groups2073611262835488442omplex @ G @ A3 )
= zero_zero_complex ) ) ).
% sum.infinite
thf(fact_1091_sum_Oinfinite,axiom,
! [A3: set_complex,G: complex > complex] :
( ~ ( finite3207457112153483333omplex @ A3 )
=> ( ( groups7754918857620584856omplex @ G @ A3 )
= zero_zero_complex ) ) ).
% sum.infinite
thf(fact_1092_sum_Odelta_H,axiom,
! [S: set_nat,A: nat,B: nat > real] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A @ S )
=> ( ( groups6591440286371151544t_real
@ ^ [K2: nat] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_real )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S )
=> ( ( groups6591440286371151544t_real
@ ^ [K2: nat] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_real )
@ S )
= zero_zero_real ) ) ) ) ).
% sum.delta'
thf(fact_1093_sum_Odelta_H,axiom,
! [S: set_complex,A: complex,B: complex > real] :
( ( finite3207457112153483333omplex @ S )
=> ( ( ( member_complex @ A @ S )
=> ( ( groups5808333547571424918x_real
@ ^ [K2: complex] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_real )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_complex @ A @ S )
=> ( ( groups5808333547571424918x_real
@ ^ [K2: complex] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_real )
@ S )
= zero_zero_real ) ) ) ) ).
% sum.delta'
thf(fact_1094_sum_Odelta_H,axiom,
! [S: set_nat,A: nat,B: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A @ S )
=> ( ( groups3542108847815614940at_nat
@ ^ [K2: nat] : ( if_nat @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_nat )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S )
=> ( ( groups3542108847815614940at_nat
@ ^ [K2: nat] : ( if_nat @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_nat )
@ S )
= zero_zero_nat ) ) ) ) ).
% sum.delta'
thf(fact_1095_sum_Odelta_H,axiom,
! [S: set_complex,A: complex,B: complex > nat] :
( ( finite3207457112153483333omplex @ S )
=> ( ( ( member_complex @ A @ S )
=> ( ( groups5693394587270226106ex_nat
@ ^ [K2: complex] : ( if_nat @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_nat )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_complex @ A @ S )
=> ( ( groups5693394587270226106ex_nat
@ ^ [K2: complex] : ( if_nat @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_nat )
@ S )
= zero_zero_nat ) ) ) ) ).
% sum.delta'
thf(fact_1096_sum_Odelta_H,axiom,
! [S: set_nat,A: nat,B: nat > complex] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A @ S )
=> ( ( groups2073611262835488442omplex
@ ^ [K2: nat] : ( if_complex @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_complex )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S )
=> ( ( groups2073611262835488442omplex
@ ^ [K2: nat] : ( if_complex @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_complex )
@ S )
= zero_zero_complex ) ) ) ) ).
% sum.delta'
thf(fact_1097_sum_Odelta_H,axiom,
! [S: set_complex,A: complex,B: complex > complex] :
( ( finite3207457112153483333omplex @ S )
=> ( ( ( member_complex @ A @ S )
=> ( ( groups7754918857620584856omplex
@ ^ [K2: complex] : ( if_complex @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_complex )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_complex @ A @ S )
=> ( ( groups7754918857620584856omplex
@ ^ [K2: complex] : ( if_complex @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_complex )
@ S )
= zero_zero_complex ) ) ) ) ).
% sum.delta'
thf(fact_1098_sum_Odelta,axiom,
! [S: set_nat,A: nat,B: nat > real] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A @ S )
=> ( ( groups6591440286371151544t_real
@ ^ [K2: nat] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_real )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S )
=> ( ( groups6591440286371151544t_real
@ ^ [K2: nat] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_real )
@ S )
= zero_zero_real ) ) ) ) ).
% sum.delta
thf(fact_1099_sum_Odelta,axiom,
! [S: set_complex,A: complex,B: complex > real] :
( ( finite3207457112153483333omplex @ S )
=> ( ( ( member_complex @ A @ S )
=> ( ( groups5808333547571424918x_real
@ ^ [K2: complex] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_real )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_complex @ A @ S )
=> ( ( groups5808333547571424918x_real
@ ^ [K2: complex] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_real )
@ S )
= zero_zero_real ) ) ) ) ).
% sum.delta
thf(fact_1100_sum_Odelta,axiom,
! [S: set_nat,A: nat,B: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A @ S )
=> ( ( groups3542108847815614940at_nat
@ ^ [K2: nat] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_nat )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S )
=> ( ( groups3542108847815614940at_nat
@ ^ [K2: nat] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_nat )
@ S )
= zero_zero_nat ) ) ) ) ).
% sum.delta
thf(fact_1101_sum_Odelta,axiom,
! [S: set_complex,A: complex,B: complex > nat] :
( ( finite3207457112153483333omplex @ S )
=> ( ( ( member_complex @ A @ S )
=> ( ( groups5693394587270226106ex_nat
@ ^ [K2: complex] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_nat )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_complex @ A @ S )
=> ( ( groups5693394587270226106ex_nat
@ ^ [K2: complex] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_nat )
@ S )
= zero_zero_nat ) ) ) ) ).
% sum.delta
thf(fact_1102_sum_Odelta,axiom,
! [S: set_nat,A: nat,B: nat > complex] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A @ S )
=> ( ( groups2073611262835488442omplex
@ ^ [K2: nat] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_complex )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S )
=> ( ( groups2073611262835488442omplex
@ ^ [K2: nat] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_complex )
@ S )
= zero_zero_complex ) ) ) ) ).
% sum.delta
thf(fact_1103_sum_Odelta,axiom,
! [S: set_complex,A: complex,B: complex > complex] :
( ( finite3207457112153483333omplex @ S )
=> ( ( ( member_complex @ A @ S )
=> ( ( groups7754918857620584856omplex
@ ^ [K2: complex] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_complex )
@ S )
= ( B @ A ) ) )
& ( ~ ( member_complex @ A @ S )
=> ( ( groups7754918857620584856omplex
@ ^ [K2: complex] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_complex )
@ S )
= zero_zero_complex ) ) ) ) ).
% sum.delta
thf(fact_1104_sum__zero__power,axiom,
! [A3: set_nat,C: nat > complex] :
( ( ( ( finite_finite_nat @ A3 )
& ( member_nat @ zero_zero_nat @ A3 ) )
=> ( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ zero_zero_complex @ I3 ) )
@ A3 )
= ( C @ zero_zero_nat ) ) )
& ( ~ ( ( finite_finite_nat @ A3 )
& ( member_nat @ zero_zero_nat @ A3 ) )
=> ( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ zero_zero_complex @ I3 ) )
@ A3 )
= zero_zero_complex ) ) ) ).
% sum_zero_power
thf(fact_1105_sum__zero__power,axiom,
! [A3: set_nat,C: nat > real] :
( ( ( ( finite_finite_nat @ A3 )
& ( member_nat @ zero_zero_nat @ A3 ) )
=> ( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ zero_zero_real @ I3 ) )
@ A3 )
= ( C @ zero_zero_nat ) ) )
& ( ~ ( ( finite_finite_nat @ A3 )
& ( member_nat @ zero_zero_nat @ A3 ) )
=> ( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ zero_zero_real @ I3 ) )
@ A3 )
= zero_zero_real ) ) ) ).
% sum_zero_power
thf(fact_1106_sum_Odistrib,axiom,
! [G: complex > complex,H: complex > complex,A3: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [X4: complex] : ( plus_plus_complex @ ( G @ X4 ) @ ( H @ X4 ) )
@ A3 )
= ( plus_plus_complex @ ( groups7754918857620584856omplex @ G @ A3 ) @ ( groups7754918857620584856omplex @ H @ A3 ) ) ) ).
% sum.distrib
thf(fact_1107_sum__divide__distrib,axiom,
! [F: complex > complex,A3: set_complex,R2: complex] :
( ( divide1717551699836669952omplex @ ( groups7754918857620584856omplex @ F @ A3 ) @ R2 )
= ( groups7754918857620584856omplex
@ ^ [N2: complex] : ( divide1717551699836669952omplex @ ( F @ N2 ) @ R2 )
@ A3 ) ) ).
% sum_divide_distrib
thf(fact_1108_sum_Oswap,axiom,
! [G: complex > complex > complex,B3: set_complex,A3: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [I3: complex] : ( groups7754918857620584856omplex @ ( G @ I3 ) @ B3 )
@ A3 )
= ( groups7754918857620584856omplex
@ ^ [J2: complex] :
( groups7754918857620584856omplex
@ ^ [I3: complex] : ( G @ I3 @ J2 )
@ A3 )
@ B3 ) ) ).
% sum.swap
thf(fact_1109_sum__product,axiom,
! [F: complex > complex,A3: set_complex,G: complex > complex,B3: set_complex] :
( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A3 ) @ ( groups7754918857620584856omplex @ G @ B3 ) )
= ( groups7754918857620584856omplex
@ ^ [I3: complex] :
( groups7754918857620584856omplex
@ ^ [J2: complex] : ( times_times_complex @ ( F @ I3 ) @ ( G @ J2 ) )
@ B3 )
@ A3 ) ) ).
% sum_product
thf(fact_1110_sum__distrib__right,axiom,
! [F: complex > complex,A3: set_complex,R2: complex] :
( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A3 ) @ R2 )
= ( groups7754918857620584856omplex
@ ^ [N2: complex] : ( times_times_complex @ ( F @ N2 ) @ R2 )
@ A3 ) ) ).
% sum_distrib_right
thf(fact_1111_sum__distrib__left,axiom,
! [R2: complex,F: complex > complex,A3: set_complex] :
( ( times_times_complex @ R2 @ ( groups7754918857620584856omplex @ F @ A3 ) )
= ( groups7754918857620584856omplex
@ ^ [N2: complex] : ( times_times_complex @ R2 @ ( F @ N2 ) )
@ A3 ) ) ).
% sum_distrib_left
thf(fact_1112_sum_Oneutral,axiom,
! [A3: set_complex,G: complex > complex] :
( ! [X2: complex] :
( ( member_complex @ X2 @ A3 )
=> ( ( G @ X2 )
= zero_zero_complex ) )
=> ( ( groups7754918857620584856omplex @ G @ A3 )
= zero_zero_complex ) ) ).
% sum.neutral
thf(fact_1113_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: complex > complex,A3: set_complex] :
( ( ( groups7754918857620584856omplex @ G @ A3 )
!= zero_zero_complex )
=> ~ ! [A5: complex] :
( ( member_complex @ A5 @ A3 )
=> ( ( G @ A5 )
= zero_zero_complex ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_1114_sum_Ointer__filter,axiom,
! [A3: set_nat,G: nat > real,P: nat > $o] :
( ( finite_finite_nat @ A3 )
=> ( ( groups6591440286371151544t_real @ G
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A3 )
& ( P @ X4 ) ) ) )
= ( groups6591440286371151544t_real
@ ^ [X4: nat] : ( if_real @ ( P @ X4 ) @ ( G @ X4 ) @ zero_zero_real )
@ A3 ) ) ) ).
% sum.inter_filter
thf(fact_1115_sum_Ointer__filter,axiom,
! [A3: set_complex,G: complex > real,P: complex > $o] :
( ( finite3207457112153483333omplex @ A3 )
=> ( ( groups5808333547571424918x_real @ G
@ ( collect_complex
@ ^ [X4: complex] :
( ( member_complex @ X4 @ A3 )
& ( P @ X4 ) ) ) )
= ( groups5808333547571424918x_real
@ ^ [X4: complex] : ( if_real @ ( P @ X4 ) @ ( G @ X4 ) @ zero_zero_real )
@ A3 ) ) ) ).
% sum.inter_filter
thf(fact_1116_sum_Ointer__filter,axiom,
! [A3: set_nat,G: nat > nat,P: nat > $o] :
( ( finite_finite_nat @ A3 )
=> ( ( groups3542108847815614940at_nat @ G
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A3 )
& ( P @ X4 ) ) ) )
= ( groups3542108847815614940at_nat
@ ^ [X4: nat] : ( if_nat @ ( P @ X4 ) @ ( G @ X4 ) @ zero_zero_nat )
@ A3 ) ) ) ).
% sum.inter_filter
thf(fact_1117_sum_Ointer__filter,axiom,
! [A3: set_complex,G: complex > nat,P: complex > $o] :
( ( finite3207457112153483333omplex @ A3 )
=> ( ( groups5693394587270226106ex_nat @ G
@ ( collect_complex
@ ^ [X4: complex] :
( ( member_complex @ X4 @ A3 )
& ( P @ X4 ) ) ) )
= ( groups5693394587270226106ex_nat
@ ^ [X4: complex] : ( if_nat @ ( P @ X4 ) @ ( G @ X4 ) @ zero_zero_nat )
@ A3 ) ) ) ).
% sum.inter_filter
thf(fact_1118_sum_Ointer__filter,axiom,
! [A3: set_nat,G: nat > complex,P: nat > $o] :
( ( finite_finite_nat @ A3 )
=> ( ( groups2073611262835488442omplex @ G
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A3 )
& ( P @ X4 ) ) ) )
= ( groups2073611262835488442omplex
@ ^ [X4: nat] : ( if_complex @ ( P @ X4 ) @ ( G @ X4 ) @ zero_zero_complex )
@ A3 ) ) ) ).
% sum.inter_filter
thf(fact_1119_sum_Ointer__filter,axiom,
! [A3: set_complex,G: complex > complex,P: complex > $o] :
( ( finite3207457112153483333omplex @ A3 )
=> ( ( groups7754918857620584856omplex @ G
@ ( collect_complex
@ ^ [X4: complex] :
( ( member_complex @ X4 @ A3 )
& ( P @ X4 ) ) ) )
= ( groups7754918857620584856omplex
@ ^ [X4: complex] : ( if_complex @ ( P @ X4 ) @ ( G @ X4 ) @ zero_zero_complex )
@ A3 ) ) ) ).
% sum.inter_filter
thf(fact_1120_sum_Oswap__restrict,axiom,
! [A3: set_nat,B3: set_complex,G: nat > complex > complex,R4: nat > complex > $o] :
( ( finite_finite_nat @ A3 )
=> ( ( finite3207457112153483333omplex @ B3 )
=> ( ( groups2073611262835488442omplex
@ ^ [X4: nat] :
( groups7754918857620584856omplex @ ( G @ X4 )
@ ( collect_complex
@ ^ [Y2: complex] :
( ( member_complex @ Y2 @ B3 )
& ( R4 @ X4 @ Y2 ) ) ) )
@ A3 )
= ( groups7754918857620584856omplex
@ ^ [Y2: complex] :
( groups2073611262835488442omplex
@ ^ [X4: nat] : ( G @ X4 @ Y2 )
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A3 )
& ( R4 @ X4 @ Y2 ) ) ) )
@ B3 ) ) ) ) ).
% sum.swap_restrict
thf(fact_1121_sum_Oswap__restrict,axiom,
! [A3: set_complex,B3: set_nat,G: complex > nat > complex,R4: complex > nat > $o] :
( ( finite3207457112153483333omplex @ A3 )
=> ( ( finite_finite_nat @ B3 )
=> ( ( groups7754918857620584856omplex
@ ^ [X4: complex] :
( groups2073611262835488442omplex @ ( G @ X4 )
@ ( collect_nat
@ ^ [Y2: nat] :
( ( member_nat @ Y2 @ B3 )
& ( R4 @ X4 @ Y2 ) ) ) )
@ A3 )
= ( groups2073611262835488442omplex
@ ^ [Y2: nat] :
( groups7754918857620584856omplex
@ ^ [X4: complex] : ( G @ X4 @ Y2 )
@ ( collect_complex
@ ^ [X4: complex] :
( ( member_complex @ X4 @ A3 )
& ( R4 @ X4 @ Y2 ) ) ) )
@ B3 ) ) ) ) ).
% sum.swap_restrict
thf(fact_1122_sum_Oswap__restrict,axiom,
! [A3: set_complex,B3: set_complex,G: complex > complex > complex,R4: complex > complex > $o] :
( ( finite3207457112153483333omplex @ A3 )
=> ( ( finite3207457112153483333omplex @ B3 )
=> ( ( groups7754918857620584856omplex
@ ^ [X4: complex] :
( groups7754918857620584856omplex @ ( G @ X4 )
@ ( collect_complex
@ ^ [Y2: complex] :
( ( member_complex @ Y2 @ B3 )
& ( R4 @ X4 @ Y2 ) ) ) )
@ A3 )
= ( groups7754918857620584856omplex
@ ^ [Y2: complex] :
( groups7754918857620584856omplex
@ ^ [X4: complex] : ( G @ X4 @ Y2 )
@ ( collect_complex
@ ^ [X4: complex] :
( ( member_complex @ X4 @ A3 )
& ( R4 @ X4 @ Y2 ) ) ) )
@ B3 ) ) ) ) ).
% sum.swap_restrict
thf(fact_1123_sum__le__included,axiom,
! [S2: set_nat,T: set_nat,G: nat > real,I: nat > nat,F: nat > real] :
( ( finite_finite_nat @ S2 )
=> ( ( finite_finite_nat @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ T )
=> ( ord_less_eq_real @ zero_zero_real @ ( G @ X2 ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S2 )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ T )
& ( ( I @ Xa2 )
= X2 )
& ( ord_less_eq_real @ ( F @ X2 ) @ ( G @ Xa2 ) ) ) )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ S2 ) @ ( groups6591440286371151544t_real @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_1124_sum__le__included,axiom,
! [S2: set_nat,T: set_complex,G: complex > real,I: complex > nat,F: nat > real] :
( ( finite_finite_nat @ S2 )
=> ( ( finite3207457112153483333omplex @ T )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ T )
=> ( ord_less_eq_real @ zero_zero_real @ ( G @ X2 ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S2 )
=> ? [Xa2: complex] :
( ( member_complex @ Xa2 @ T )
& ( ( I @ Xa2 )
= X2 )
& ( ord_less_eq_real @ ( F @ X2 ) @ ( G @ Xa2 ) ) ) )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ S2 ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_1125_sum__le__included,axiom,
! [S2: set_complex,T: set_nat,G: nat > real,I: nat > complex,F: complex > real] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( finite_finite_nat @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ T )
=> ( ord_less_eq_real @ zero_zero_real @ ( G @ X2 ) ) )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ S2 )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ T )
& ( ( I @ Xa2 )
= X2 )
& ( ord_less_eq_real @ ( F @ X2 ) @ ( G @ Xa2 ) ) ) )
=> ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S2 ) @ ( groups6591440286371151544t_real @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_1126_sum__le__included,axiom,
! [S2: set_complex,T: set_complex,G: complex > real,I: complex > complex,F: complex > real] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( finite3207457112153483333omplex @ T )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ T )
=> ( ord_less_eq_real @ zero_zero_real @ ( G @ X2 ) ) )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ S2 )
=> ? [Xa2: complex] :
( ( member_complex @ Xa2 @ T )
& ( ( I @ Xa2 )
= X2 )
& ( ord_less_eq_real @ ( F @ X2 ) @ ( G @ Xa2 ) ) ) )
=> ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S2 ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_1127_sum__le__included,axiom,
! [S2: set_nat,T: set_nat,G: nat > nat,I: nat > nat,F: nat > nat] :
( ( finite_finite_nat @ S2 )
=> ( ( finite_finite_nat @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ T )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( G @ X2 ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S2 )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ T )
& ( ( I @ Xa2 )
= X2 )
& ( ord_less_eq_nat @ ( F @ X2 ) @ ( G @ Xa2 ) ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ S2 ) @ ( groups3542108847815614940at_nat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_1128_sum__le__included,axiom,
! [S2: set_nat,T: set_complex,G: complex > nat,I: complex > nat,F: nat > nat] :
( ( finite_finite_nat @ S2 )
=> ( ( finite3207457112153483333omplex @ T )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ T )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( G @ X2 ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S2 )
=> ? [Xa2: complex] :
( ( member_complex @ Xa2 @ T )
& ( ( I @ Xa2 )
= X2 )
& ( ord_less_eq_nat @ ( F @ X2 ) @ ( G @ Xa2 ) ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ S2 ) @ ( groups5693394587270226106ex_nat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_1129_sum__le__included,axiom,
! [S2: set_complex,T: set_nat,G: nat > nat,I: nat > complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( finite_finite_nat @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ T )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( G @ X2 ) ) )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ S2 )
=> ? [Xa2: nat] :
( ( member_nat @ Xa2 @ T )
& ( ( I @ Xa2 )
= X2 )
& ( ord_less_eq_nat @ ( F @ X2 ) @ ( G @ Xa2 ) ) ) )
=> ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ S2 ) @ ( groups3542108847815614940at_nat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_1130_sum__le__included,axiom,
! [S2: set_complex,T: set_complex,G: complex > nat,I: complex > complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( finite3207457112153483333omplex @ T )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ T )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( G @ X2 ) ) )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ S2 )
=> ? [Xa2: complex] :
( ( member_complex @ Xa2 @ T )
& ( ( I @ Xa2 )
= X2 )
& ( ord_less_eq_nat @ ( F @ X2 ) @ ( G @ Xa2 ) ) ) )
=> ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ S2 ) @ ( groups5693394587270226106ex_nat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_1131_sum__nonneg__eq__0__iff,axiom,
! [A3: set_nat,F: nat > real] :
( ( finite_finite_nat @ A3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A3 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X2 ) ) )
=> ( ( ( groups6591440286371151544t_real @ F @ A3 )
= zero_zero_real )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A3 )
=> ( ( F @ X4 )
= zero_zero_real ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_1132_sum__nonneg__eq__0__iff,axiom,
! [A3: set_complex,F: complex > real] :
( ( finite3207457112153483333omplex @ A3 )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ A3 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X2 ) ) )
=> ( ( ( groups5808333547571424918x_real @ F @ A3 )
= zero_zero_real )
= ( ! [X4: complex] :
( ( member_complex @ X4 @ A3 )
=> ( ( F @ X4 )
= zero_zero_real ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_1133_sum__nonneg__eq__0__iff,axiom,
! [A3: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) ) )
=> ( ( ( groups3542108847815614940at_nat @ F @ A3 )
= zero_zero_nat )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A3 )
=> ( ( F @ X4 )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_1134_sum__nonneg__eq__0__iff,axiom,
! [A3: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ A3 )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ A3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) ) )
=> ( ( ( groups5693394587270226106ex_nat @ F @ A3 )
= zero_zero_nat )
= ( ! [X4: complex] :
( ( member_complex @ X4 @ A3 )
=> ( ( F @ X4 )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_1135_sum_Orelated,axiom,
! [R4: real > real > $o,S: set_nat,H: nat > real,G: nat > real] :
( ( R4 @ zero_zero_real @ zero_zero_real )
=> ( ! [X12: real,Y12: real,X23: real,Y23: real] :
( ( ( R4 @ X12 @ X23 )
& ( R4 @ Y12 @ Y23 ) )
=> ( R4 @ ( plus_plus_real @ X12 @ Y12 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S )
=> ( R4 @ ( H @ X2 ) @ ( G @ X2 ) ) )
=> ( R4 @ ( groups6591440286371151544t_real @ H @ S ) @ ( groups6591440286371151544t_real @ G @ S ) ) ) ) ) ) ).
% sum.related
thf(fact_1136_sum_Orelated,axiom,
! [R4: real > real > $o,S: set_complex,H: complex > real,G: complex > real] :
( ( R4 @ zero_zero_real @ zero_zero_real )
=> ( ! [X12: real,Y12: real,X23: real,Y23: real] :
( ( ( R4 @ X12 @ X23 )
& ( R4 @ Y12 @ Y23 ) )
=> ( R4 @ ( plus_plus_real @ X12 @ Y12 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
=> ( ( finite3207457112153483333omplex @ S )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ S )
=> ( R4 @ ( H @ X2 ) @ ( G @ X2 ) ) )
=> ( R4 @ ( groups5808333547571424918x_real @ H @ S ) @ ( groups5808333547571424918x_real @ G @ S ) ) ) ) ) ) ).
% sum.related
thf(fact_1137_sum_Orelated,axiom,
! [R4: nat > nat > $o,S: set_nat,H: nat > nat,G: nat > nat] :
( ( R4 @ zero_zero_nat @ zero_zero_nat )
=> ( ! [X12: nat,Y12: nat,X23: nat,Y23: nat] :
( ( ( R4 @ X12 @ X23 )
& ( R4 @ Y12 @ Y23 ) )
=> ( R4 @ ( plus_plus_nat @ X12 @ Y12 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S )
=> ( R4 @ ( H @ X2 ) @ ( G @ X2 ) ) )
=> ( R4 @ ( groups3542108847815614940at_nat @ H @ S ) @ ( groups3542108847815614940at_nat @ G @ S ) ) ) ) ) ) ).
% sum.related
thf(fact_1138_sum_Orelated,axiom,
! [R4: nat > nat > $o,S: set_complex,H: complex > nat,G: complex > nat] :
( ( R4 @ zero_zero_nat @ zero_zero_nat )
=> ( ! [X12: nat,Y12: nat,X23: nat,Y23: nat] :
( ( ( R4 @ X12 @ X23 )
& ( R4 @ Y12 @ Y23 ) )
=> ( R4 @ ( plus_plus_nat @ X12 @ Y12 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
=> ( ( finite3207457112153483333omplex @ S )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ S )
=> ( R4 @ ( H @ X2 ) @ ( G @ X2 ) ) )
=> ( R4 @ ( groups5693394587270226106ex_nat @ H @ S ) @ ( groups5693394587270226106ex_nat @ G @ S ) ) ) ) ) ) ).
% sum.related
thf(fact_1139_sum_Orelated,axiom,
! [R4: complex > complex > $o,S: set_nat,H: nat > complex,G: nat > complex] :
( ( R4 @ zero_zero_complex @ zero_zero_complex )
=> ( ! [X12: complex,Y12: complex,X23: complex,Y23: complex] :
( ( ( R4 @ X12 @ X23 )
& ( R4 @ Y12 @ Y23 ) )
=> ( R4 @ ( plus_plus_complex @ X12 @ Y12 ) @ ( plus_plus_complex @ X23 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S )
=> ( R4 @ ( H @ X2 ) @ ( G @ X2 ) ) )
=> ( R4 @ ( groups2073611262835488442omplex @ H @ S ) @ ( groups2073611262835488442omplex @ G @ S ) ) ) ) ) ) ).
% sum.related
thf(fact_1140_sum_Orelated,axiom,
! [R4: complex > complex > $o,S: set_complex,H: complex > complex,G: complex > complex] :
( ( R4 @ zero_zero_complex @ zero_zero_complex )
=> ( ! [X12: complex,Y12: complex,X23: complex,Y23: complex] :
( ( ( R4 @ X12 @ X23 )
& ( R4 @ Y12 @ Y23 ) )
=> ( R4 @ ( plus_plus_complex @ X12 @ Y12 ) @ ( plus_plus_complex @ X23 @ Y23 ) ) )
=> ( ( finite3207457112153483333omplex @ S )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ S )
=> ( R4 @ ( H @ X2 ) @ ( G @ X2 ) ) )
=> ( R4 @ ( groups7754918857620584856omplex @ H @ S ) @ ( groups7754918857620584856omplex @ G @ S ) ) ) ) ) ) ).
% sum.related
thf(fact_1141_sum__nonneg__0,axiom,
! [S2: set_nat,F: nat > real,I: nat] :
( ( finite_finite_nat @ S2 )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ S2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
=> ( ( ( groups6591440286371151544t_real @ F @ S2 )
= zero_zero_real )
=> ( ( member_nat @ I @ S2 )
=> ( ( F @ I )
= zero_zero_real ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_1142_sum__nonneg__0,axiom,
! [S2: set_complex,F: complex > real,I: complex] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ! [I4: complex] :
( ( member_complex @ I4 @ S2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
=> ( ( ( groups5808333547571424918x_real @ F @ S2 )
= zero_zero_real )
=> ( ( member_complex @ I @ S2 )
=> ( ( F @ I )
= zero_zero_real ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_1143_sum__nonneg__0,axiom,
! [S2: set_nat,F: nat > nat,I: nat] :
( ( finite_finite_nat @ S2 )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ S2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
=> ( ( ( groups3542108847815614940at_nat @ F @ S2 )
= zero_zero_nat )
=> ( ( member_nat @ I @ S2 )
=> ( ( F @ I )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_1144_sum__nonneg__0,axiom,
! [S2: set_complex,F: complex > nat,I: complex] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ! [I4: complex] :
( ( member_complex @ I4 @ S2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
=> ( ( ( groups5693394587270226106ex_nat @ F @ S2 )
= zero_zero_nat )
=> ( ( member_complex @ I @ S2 )
=> ( ( F @ I )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_1145_sum__nonneg__leq__bound,axiom,
! [S2: set_nat,F: nat > real,B3: real,I: nat] :
( ( finite_finite_nat @ S2 )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ S2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
=> ( ( ( groups6591440286371151544t_real @ F @ S2 )
= B3 )
=> ( ( member_nat @ I @ S2 )
=> ( ord_less_eq_real @ ( F @ I ) @ B3 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_1146_sum__nonneg__leq__bound,axiom,
! [S2: set_complex,F: complex > real,B3: real,I: complex] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ! [I4: complex] :
( ( member_complex @ I4 @ S2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
=> ( ( ( groups5808333547571424918x_real @ F @ S2 )
= B3 )
=> ( ( member_complex @ I @ S2 )
=> ( ord_less_eq_real @ ( F @ I ) @ B3 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_1147_sum__nonneg__leq__bound,axiom,
! [S2: set_nat,F: nat > nat,B3: nat,I: nat] :
( ( finite_finite_nat @ S2 )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ S2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
=> ( ( ( groups3542108847815614940at_nat @ F @ S2 )
= B3 )
=> ( ( member_nat @ I @ S2 )
=> ( ord_less_eq_nat @ ( F @ I ) @ B3 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_1148_sum__nonneg__leq__bound,axiom,
! [S2: set_complex,F: complex > nat,B3: nat,I: complex] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ! [I4: complex] :
( ( member_complex @ I4 @ S2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
=> ( ( ( groups5693394587270226106ex_nat @ F @ S2 )
= B3 )
=> ( ( member_complex @ I @ S2 )
=> ( ord_less_eq_nat @ ( F @ I ) @ B3 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_1149_sum__power__add,axiom,
! [X3: complex,M: nat,I2: set_nat] :
( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( power_power_complex @ X3 @ ( plus_plus_nat @ M @ I3 ) )
@ I2 )
= ( times_times_complex @ ( power_power_complex @ X3 @ M ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ I2 ) ) ) ).
% sum_power_add
thf(fact_1150_sum__power__add,axiom,
! [X3: real,M: nat,I2: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( power_power_real @ X3 @ ( plus_plus_nat @ M @ I3 ) )
@ I2 )
= ( times_times_real @ ( power_power_real @ X3 @ M ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ I2 ) ) ) ).
% sum_power_add
thf(fact_1151_sum__pos2,axiom,
! [I2: set_nat,I: nat,F: nat > real] :
( ( finite_finite_nat @ I2 )
=> ( ( member_nat @ I @ I2 )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
=> ( ord_less_real @ zero_zero_real @ ( groups6591440286371151544t_real @ F @ I2 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_1152_sum__pos2,axiom,
! [I2: set_complex,I: complex,F: complex > real] :
( ( finite3207457112153483333omplex @ I2 )
=> ( ( member_complex @ I @ I2 )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
=> ( ! [I4: complex] :
( ( member_complex @ I4 @ I2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
=> ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I2 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_1153_sum__pos2,axiom,
! [I2: set_nat,I: nat,F: nat > nat] :
( ( finite_finite_nat @ I2 )
=> ( ( member_nat @ I @ I2 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups3542108847815614940at_nat @ F @ I2 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_1154_sum__pos2,axiom,
! [I2: set_complex,I: complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ I2 )
=> ( ( member_complex @ I @ I2 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
=> ( ! [I4: complex] :
( ( member_complex @ I4 @ I2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ I2 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_1155_sum__nth__roots,axiom,
! [N: nat,C: complex] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( groups7754918857620584856omplex
@ ^ [X4: complex] : X4
@ ( collect_complex
@ ^ [Z4: complex] :
( ( power_power_complex @ Z4 @ N )
= C ) ) )
= zero_zero_complex ) ) ).
% sum_nth_roots
thf(fact_1156_sum__roots__unity,axiom,
! [N: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( groups7754918857620584856omplex
@ ^ [X4: complex] : X4
@ ( collect_complex
@ ^ [Z4: complex] :
( ( power_power_complex @ Z4 @ N )
= one_one_complex ) ) )
= zero_zero_complex ) ) ).
% sum_roots_unity
thf(fact_1157_num__relations_Oelims,axiom,
! [X3: joinTree_a,Y: nat] :
( ( ( num_relations_a @ X3 )
= Y )
=> ( ( ? [Uu: a] :
( X3
= ( relation_a @ Uu ) )
=> ( Y != one_one_nat ) )
=> ~ ! [L2: joinTree_a,R: joinTree_a] :
( ( X3
= ( join_a @ L2 @ R ) )
=> ( Y
!= ( plus_plus_nat @ ( num_relations_a @ L2 ) @ ( num_relations_a @ R ) ) ) ) ) ) ).
% num_relations.elims
thf(fact_1158_polyfun__finite__roots,axiom,
! [C: nat > complex,N: nat] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X4: complex] :
( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ X4 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_complex ) ) )
= ( ? [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N )
& ( ( C @ I3 )
!= zero_zero_complex ) ) ) ) ).
% polyfun_finite_roots
thf(fact_1159_polyfun__finite__roots,axiom,
! [C: nat > real,N: nat] :
( ( finite_finite_real
@ ( collect_real
@ ^ [X4: real] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ X4 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_real ) ) )
= ( ? [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N )
& ( ( C @ I3 )
!= zero_zero_real ) ) ) ) ).
% polyfun_finite_roots
thf(fact_1160_sum_OatMost__Suc,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atMost_Suc
thf(fact_1161_atMost__def,axiom,
( set_ord_atMost_real
= ( ^ [U2: real] :
( collect_real
@ ^ [X4: real] : ( ord_less_eq_real @ X4 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_1162_atMost__def,axiom,
( set_ord_atMost_nat
= ( ^ [U2: nat] :
( collect_nat
@ ^ [X4: nat] : ( ord_less_eq_nat @ X4 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_1163_num__relations_Osimps_I2_J,axiom,
! [L: joinTree_a,R2: joinTree_a] :
( ( num_relations_a @ ( join_a @ L @ R2 ) )
= ( plus_plus_nat @ ( num_relations_a @ L ) @ ( num_relations_a @ R2 ) ) ) ).
% num_relations.simps(2)
thf(fact_1164_num__relations_Osimps_I1_J,axiom,
! [Uu2: a] :
( ( num_relations_a @ ( relation_a @ Uu2 ) )
= one_one_nat ) ).
% num_relations.simps(1)
thf(fact_1165_sum_OatMost__Suc__shift,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( plus_plus_real @ ( G @ zero_zero_nat )
@ ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_1166_polyfun__eq__coeffs,axiom,
! [C: nat > complex,N: nat,D: nat > complex] :
( ( ! [X4: complex] :
( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ X4 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( D @ I3 ) @ ( power_power_complex @ X4 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) )
= ( ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N )
=> ( ( C @ I3 )
= ( D @ I3 ) ) ) ) ) ).
% polyfun_eq_coeffs
thf(fact_1167_polyfun__eq__coeffs,axiom,
! [C: nat > real,N: nat,D: nat > real] :
( ( ! [X4: real] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ X4 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( D @ I3 ) @ ( power_power_real @ X4 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) )
= ( ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N )
=> ( ( C @ I3 )
= ( D @ I3 ) ) ) ) ) ).
% polyfun_eq_coeffs
thf(fact_1168_polyfun__eq__0,axiom,
! [C: nat > complex,N: nat] :
( ( ! [X4: complex] :
( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ X4 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_complex ) )
= ( ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N )
=> ( ( C @ I3 )
= zero_zero_complex ) ) ) ) ).
% polyfun_eq_0
thf(fact_1169_polyfun__eq__0,axiom,
! [C: nat > real,N: nat] :
( ( ! [X4: real] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ X4 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_real ) )
= ( ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N )
=> ( ( C @ I3 )
= zero_zero_real ) ) ) ) ).
% polyfun_eq_0
thf(fact_1170_zero__polynom__imp__zero__coeffs,axiom,
! [C: nat > complex,N: nat,K: nat] :
( ! [W2: complex] :
( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ W2 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_complex )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( C @ K )
= zero_zero_complex ) ) ) ).
% zero_polynom_imp_zero_coeffs
thf(fact_1171_zero__polynom__imp__zero__coeffs,axiom,
! [C: nat > real,N: nat,K: nat] :
( ! [W2: real] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ W2 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_real )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( C @ K )
= zero_zero_real ) ) ) ).
% zero_polynom_imp_zero_coeffs
thf(fact_1172_polyfun__roots__finite,axiom,
! [C: nat > complex,K: nat,N: nat] :
( ( ( C @ K )
!= zero_zero_complex )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [Z4: complex] :
( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ Z4 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_complex ) ) ) ) ) ).
% polyfun_roots_finite
thf(fact_1173_polyfun__roots__finite,axiom,
! [C: nat > real,K: nat,N: nat] :
( ( ( C @ K )
!= zero_zero_real )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( finite_finite_real
@ ( collect_real
@ ^ [Z4: real] :
( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ Z4 @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_real ) ) ) ) ) ).
% polyfun_roots_finite
thf(fact_1174_root__polyfun,axiom,
! [N: nat,Z: complex,A: complex] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( ( power_power_complex @ Z @ N )
= A )
= ( ( groups2073611262835488442omplex
@ ^ [I3: nat] : ( times_times_complex @ ( if_complex @ ( I3 = zero_zero_nat ) @ ( uminus1482373934393186551omplex @ A ) @ ( if_complex @ ( I3 = N ) @ one_one_complex @ zero_zero_complex ) ) @ ( power_power_complex @ Z @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_complex ) ) ) ).
% root_polyfun
thf(fact_1175_root__polyfun,axiom,
! [N: nat,Z: real,A: real] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( ( power_power_real @ Z @ N )
= A )
= ( ( groups6591440286371151544t_real
@ ^ [I3: nat] : ( times_times_real @ ( if_real @ ( I3 = zero_zero_nat ) @ ( uminus_uminus_real @ A ) @ ( if_real @ ( I3 = N ) @ one_one_real @ zero_zero_real ) ) @ ( power_power_real @ Z @ I3 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_real ) ) ) ).
% root_polyfun
thf(fact_1176_sum_Ozero__middle,axiom,
! [P2: nat,K: nat,G: nat > real,H: nat > real] :
( ( ord_less_eq_nat @ one_one_nat @ P2 )
=> ( ( ord_less_eq_nat @ K @ P2 )
=> ( ( groups6591440286371151544t_real
@ ^ [J2: nat] : ( if_real @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( if_real @ ( J2 = K ) @ zero_zero_real @ ( H @ ( minus_minus_nat @ J2 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P2 ) )
= ( groups6591440286371151544t_real
@ ^ [J2: nat] : ( if_real @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( H @ J2 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_1177_sum_Ozero__middle,axiom,
! [P2: nat,K: nat,G: nat > nat,H: nat > nat] :
( ( ord_less_eq_nat @ one_one_nat @ P2 )
=> ( ( ord_less_eq_nat @ K @ P2 )
=> ( ( groups3542108847815614940at_nat
@ ^ [J2: nat] : ( if_nat @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( if_nat @ ( J2 = K ) @ zero_zero_nat @ ( H @ ( minus_minus_nat @ J2 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P2 ) )
= ( groups3542108847815614940at_nat
@ ^ [J2: nat] : ( if_nat @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( H @ J2 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_1178_sum_Ozero__middle,axiom,
! [P2: nat,K: nat,G: nat > complex,H: nat > complex] :
( ( ord_less_eq_nat @ one_one_nat @ P2 )
=> ( ( ord_less_eq_nat @ K @ P2 )
=> ( ( groups2073611262835488442omplex
@ ^ [J2: nat] : ( if_complex @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( if_complex @ ( J2 = K ) @ zero_zero_complex @ ( H @ ( minus_minus_nat @ J2 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P2 ) )
= ( groups2073611262835488442omplex
@ ^ [J2: nat] : ( if_complex @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( H @ J2 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_1179_add_Oinverse__inverse,axiom,
! [A: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_1180_neg__equal__iff__equal,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= ( uminus_uminus_real @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_1181_verit__minus__simplify_I4_J,axiom,
! [B: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_1182_diff__self,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% diff_self
thf(fact_1183_diff__self,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ A )
= zero_zero_complex ) ).
% diff_self
thf(fact_1184_diff__0__right,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_0_right
thf(fact_1185_diff__0__right,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ zero_zero_complex )
= A ) ).
% diff_0_right
thf(fact_1186_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_1187_diff__zero,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_zero
thf(fact_1188_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_1189_diff__zero,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ zero_zero_complex )
= A ) ).
% diff_zero
thf(fact_1190_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1191_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1192_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ A )
= zero_zero_complex ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1193_neg__le__iff__le,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_1194_add_Oinverse__neutral,axiom,
( ( uminus1482373934393186551omplex @ zero_zero_complex )
= zero_zero_complex ) ).
% add.inverse_neutral
thf(fact_1195_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_1196_neg__0__equal__iff__equal,axiom,
! [A: complex] :
( ( zero_zero_complex
= ( uminus1482373934393186551omplex @ A ) )
= ( zero_zero_complex = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_1197_neg__0__equal__iff__equal,axiom,
! [A: real] :
( ( zero_zero_real
= ( uminus_uminus_real @ A ) )
= ( zero_zero_real = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_1198_neg__equal__0__iff__equal,axiom,
! [A: complex] :
( ( ( uminus1482373934393186551omplex @ A )
= zero_zero_complex )
= ( A = zero_zero_complex ) ) ).
% neg_equal_0_iff_equal
thf(fact_1199_neg__equal__0__iff__equal,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% neg_equal_0_iff_equal
thf(fact_1200_equal__neg__zero,axiom,
! [A: real] :
( ( A
= ( uminus_uminus_real @ A ) )
= ( A = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_1201_neg__equal__zero,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= A )
= ( A = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_1202_neg__less__iff__less,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_1203_add__diff__cancel,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_1204_diff__add__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_1205_add__diff__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( minus_minus_real @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_1206_add__diff__cancel__left_H,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_1207_add__diff__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( minus_minus_real @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_1208_add__diff__cancel__right_H,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_1209_mult__minus__left,axiom,
! [A: real,B: real] :
( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
= ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).
% mult_minus_left
thf(fact_1210_minus__mult__minus,axiom,
! [A: real,B: real] :
( ( times_times_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
= ( times_times_real @ A @ B ) ) ).
% minus_mult_minus
thf(fact_1211_mult__minus__right,axiom,
! [A: real,B: real] :
( ( times_times_real @ A @ ( uminus_uminus_real @ B ) )
= ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).
% mult_minus_right
thf(fact_1212_add__minus__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_1213_minus__add__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_1214_minus__add__distrib,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).
% minus_add_distrib
thf(fact_1215_minus__diff__eq,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) )
= ( minus_minus_real @ B @ A ) ) ).
% minus_diff_eq
thf(fact_1216_real__add__minus__iff,axiom,
! [X3: real,A: real] :
( ( ( plus_plus_real @ X3 @ ( uminus_uminus_real @ A ) )
= zero_zero_real )
= ( X3 = A ) ) ).
% real_add_minus_iff
thf(fact_1217_diff__ge__0__iff__ge,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_1218_diff__gt__0__iff__gt,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_real @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_1219_neg__less__eq__nonneg,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_1220_less__eq__neg__nonpos,axiom,
! [A: real] :
( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% less_eq_neg_nonpos
thf(fact_1221_neg__le__0__iff__le,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% neg_le_0_iff_le
thf(fact_1222_neg__0__le__iff__le,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% neg_0_le_iff_le
thf(fact_1223_less__neg__neg,axiom,
! [A: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% less_neg_neg
thf(fact_1224_neg__less__pos,axiom,
! [A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% neg_less_pos
thf(fact_1225_neg__0__less__iff__less,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% neg_0_less_iff_less
thf(fact_1226_neg__less__0__iff__less,axiom,
! [A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% neg_less_0_iff_less
thf(fact_1227_le__add__diff__inverse2,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_1228_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_1229_le__add__diff__inverse,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_1230_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_1231_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_1232_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_1233_diff__numeral__special_I9_J,axiom,
( ( minus_minus_complex @ one_one_complex @ one_one_complex )
= zero_zero_complex ) ).
% diff_numeral_special(9)
thf(fact_1234_ab__left__minus,axiom,
! [A: complex] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
= zero_zero_complex ) ).
% ab_left_minus
thf(fact_1235_ab__left__minus,axiom,
! [A: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
= zero_zero_real ) ).
% ab_left_minus
thf(fact_1236_add_Oright__inverse,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ ( uminus1482373934393186551omplex @ A ) )
= zero_zero_complex ) ).
% add.right_inverse
thf(fact_1237_add_Oright__inverse,axiom,
! [A: real] :
( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
= zero_zero_real ) ).
% add.right_inverse
thf(fact_1238_diff__0,axiom,
! [A: complex] :
( ( minus_minus_complex @ zero_zero_complex @ A )
= ( uminus1482373934393186551omplex @ A ) ) ).
% diff_0
thf(fact_1239_diff__0,axiom,
! [A: real] :
( ( minus_minus_real @ zero_zero_real @ A )
= ( uminus_uminus_real @ A ) ) ).
% diff_0
thf(fact_1240_verit__minus__simplify_I3_J,axiom,
! [B: complex] :
( ( minus_minus_complex @ zero_zero_complex @ B )
= ( uminus1482373934393186551omplex @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_1241_verit__minus__simplify_I3_J,axiom,
! [B: real] :
( ( minus_minus_real @ zero_zero_real @ B )
= ( uminus_uminus_real @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_1242_mult__minus1,axiom,
! [Z: complex] :
( ( times_times_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ Z )
= ( uminus1482373934393186551omplex @ Z ) ) ).
% mult_minus1
thf(fact_1243_mult__minus1,axiom,
! [Z: real] :
( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z )
= ( uminus_uminus_real @ Z ) ) ).
% mult_minus1
thf(fact_1244_mult__minus1__right,axiom,
! [Z: complex] :
( ( times_times_complex @ Z @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( uminus1482373934393186551omplex @ Z ) ) ).
% mult_minus1_right
thf(fact_1245_mult__minus1__right,axiom,
! [Z: real] :
( ( times_times_real @ Z @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ Z ) ) ).
% mult_minus1_right
thf(fact_1246_uminus__add__conv__diff,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B )
= ( minus_minus_real @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_1247_diff__minus__eq__add,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ A @ ( uminus_uminus_real @ B ) )
= ( plus_plus_real @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_1248_dbl__inc__simps_I4_J,axiom,
( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% dbl_inc_simps(4)
thf(fact_1249_dbl__inc__simps_I4_J,axiom,
( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ one_one_real ) ) ).
% dbl_inc_simps(4)
thf(fact_1250_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
= zero_zero_complex ) ).
% add_neg_numeral_special(8)
thf(fact_1251_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
= zero_zero_real ) ).
% add_neg_numeral_special(8)
thf(fact_1252_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= zero_zero_complex ) ).
% add_neg_numeral_special(7)
thf(fact_1253_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
= zero_zero_real ) ).
% add_neg_numeral_special(7)
thf(fact_1254_diff__numeral__special_I12_J,axiom,
( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= zero_zero_complex ) ).
% diff_numeral_special(12)
thf(fact_1255_diff__numeral__special_I12_J,axiom,
( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
= zero_zero_real ) ).
% diff_numeral_special(12)
thf(fact_1256_minus__one__mult__self,axiom,
! [N: nat] :
( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) )
= one_one_complex ) ).
% minus_one_mult_self
thf(fact_1257_minus__one__mult__self,axiom,
! [N: nat] :
( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) )
= one_one_real ) ).
% minus_one_mult_self
thf(fact_1258_left__minus__one__mult__self,axiom,
! [N: nat,A: complex] :
( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ A ) )
= A ) ).
% left_minus_one_mult_self
thf(fact_1259_left__minus__one__mult__self,axiom,
! [N: nat,A: real] :
( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ A ) )
= A ) ).
% left_minus_one_mult_self
thf(fact_1260_sum__negf,axiom,
! [F: complex > complex,A3: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [X4: complex] : ( uminus1482373934393186551omplex @ ( F @ X4 ) )
@ A3 )
= ( uminus1482373934393186551omplex @ ( groups7754918857620584856omplex @ F @ A3 ) ) ) ).
% sum_negf
thf(fact_1261_sum__subtractf,axiom,
! [F: complex > complex,G: complex > complex,A3: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [X4: complex] : ( minus_minus_complex @ ( F @ X4 ) @ ( G @ X4 ) )
@ A3 )
= ( minus_minus_complex @ ( groups7754918857620584856omplex @ F @ A3 ) @ ( groups7754918857620584856omplex @ G @ A3 ) ) ) ).
% sum_subtractf
thf(fact_1262_power__minus,axiom,
! [A: complex,N: nat] :
( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
= ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ A @ N ) ) ) ).
% power_minus
thf(fact_1263_power__minus,axiom,
! [A: real,N: nat] :
( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
= ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ A @ N ) ) ) ).
% power_minus
% Helper facts (7)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y: nat] :
( ( if_nat @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y: nat] :
( ( if_nat @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X3: real,Y: real] :
( ( if_real @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X3: real,Y: real] :
( ( if_real @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_3_1_If_001t__Complex__Ocomplex_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
! [X3: complex,Y: complex] :
( ( if_complex @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
! [X3: complex,Y: complex] :
( ( if_complex @ $true @ X3 @ Y )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( c_IKKBZ_a @ h1 @ cf @ f @ ( join_a @ l @ ( relation_a @ r ) ) )
= ( plus_plus_real @ ( times_times_real @ ( card_a @ cf @ f @ l ) @ ( h1 @ r @ ( cf @ r ) ) ) @ ( c_list_a @ ( ldeep_s_a @ f @ ( revorder_a @ l ) ) @ cf @ h2 @ ( first_node_a @ ( join_a @ l @ ( relation_a @ r ) ) ) @ ( revorder_a @ l ) ) ) ) ).
%------------------------------------------------------------------------------